IMAGING PERFORMANCE OF ELLIPTICAL-BOUNDARY VARIFOCAL MIRRORS IN ACTIVE OPTICAL SYSTEMS by

advertisement

IMAGING PERFORMANCE OF ELLIPTICAL-BOUNDARY VARIFOCAL

MIRRORS IN ACTIVE OPTICAL SYSTEMS by

Sarah Jane Lukes

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in

Engineering

MONTANA STATE UNIVERSITY

Bozeman, Montana

January 2015

©COPYRIGHT by

Sarah Jane Lukes

2015

All Rights Reserved

ii

DEDICATION

To Bev –

May you have experienced fair pay and the safety, food, shelter, reliable transportation, healthcare, security, and peace of mind that it provides. Your graceful strength carried me through the more difficult times in this endeavor.

iii

ACKNOWLEDGEMENTS

I am most grateful to have the love and support of my husband. Thanks for providing a daily dose of humor and for being a major contributor on the home front, even if it means our kid is frequently dressed in a striped pajama top with camo pants.

Thanks to my son for enriching my life and broadening my perspective. I am grateful for my parents’ sacrifices. I know if Mom were here today she would say, “Oh honey, I am so proud of you. Now, what exactly did you study again?” A special thanks to Erin Casey for providing encouragement to finish when I needed encouragement most. I am deeply appreciative of the healing comfort of our friends who graced us with their company and warm meals over the years, especially, the Daughenbaughs, Koons, Kays, Koturs, Nancy,

Janet, Erin, and the lovely ladies of Meals for Moms.

Thanks to my adviser, David Dickensheets, for his overall research vision and depth of knowledge. Thanks for allowing me freedom to make mistakes and explore different topics, encouraging me to make decisions quickly, and helping me to frame my work appropriately. Thanks to my committee members, Todd Kaiser, Charles

Kankelborg, and Joseph Shaw for their time, insight, and ease of scheduling. I appreciate those who have helped me with professional development, especially, Robert Maher,

Sarah Codd, Deanne, Emily, A.J., Katie, Kasey, Matt, Addie, Cat, and Elmira. Thanks to

Ryan for maintaining lab interferometers. He and Tesha assisted with mirror packaging.

This project was supported by the Montana Microfabrication Facility, Lurie

Nanofabrication Facility, and by NSF under projects DBI-0754608, DBI-1152631, and

IIP-0956910. Sarah Lukes was supported by an NSF Graduate Research Fellowship.

iv

TABLE OF CONTENTS

1.

INTRODUCTION ........................................................................................................ 1

History........................................................................................................................... 1

Deformable-Membrane MEMS Mirrors ................................................................ 3

Previous MEMS Mirror Development .................................................................. 7

Evaluation of Imaging Performance of MEMS Mirrors ...................................... 10

Motivation for Aberration Analysis ..................................................................... 13

Relevant Analyses of Aberrations for Varifocal Mirrors ........................................... 15

Contribution of This Work.......................................................................................... 15

Dissertation Organization ........................................................................................... 17

2.

MEMS MIRROR DEVELOPMENT ......................................................................... 18

Novel MEMS Mirror Designs .................................................................................... 18

Wet- and Dry-release MEMS Mirrors on Single Silicon Wafer ......................... 18

Dry-release MEMS Mirrors on Single SOI Wafer .............................................. 21

Device Characterization .............................................................................................. 26

Displacement due to Electrostatic Pressure ......................................................... 27

Dynamic Characterization ................................................................................... 29

Surface Shape ...................................................................................................... 29

3.

FRAMEWORK FOR HAMILTONIAN OPTICS ANALYSIS ................................. 39

Coordinate System and Aberration Coefficients ........................................................ 39

Stop Shifting ........................................................................................................ 46

Implementation Details ........................................................................................ 47

Definition of Surface Shape for Elliptical-Boundary Mirrors .................................... 49

4.

IMAGING PERFORMANCE OF A SINGLE MEMS MIRROR ............................. 51

Parabasal Coefficients ................................................................................................. 51

Second-Order Aberrations .......................................................................................... 53

Peak-to-valley, 2 nd

-Order Aberration Terms for an Object at Infinity ................ 59

Third-Order Aberrations ............................................................................................. 61

Seidel Aberration Terms for Normal-Incidence Mirror ...................................... 67 rd

Peak-to-valley 3 -order Aberration Terms for an Object at Infinity .................. 70

RMS Wavefront Aberration ........................................................................................ 83

Comparison of a Circular and Elliptical Mirror at 5 o

Incidence Angle ...................... 93

Strehl Ratio ................................................................................................................. 94

Conclusions and Guidelines for a Paraboloidal Dynamic Varifocal Mirror............... 95

Comparison of 2 nd

- and 3 rd

-order Terms ............................................................. 95

Incidence Angle and Field Limitations ................................................................ 99

Zemax Overview ............................................................................................... 101

v

TABLE OF CONTENTS – CONTINUED

One Wave of 2

One Wave of 2 nd nd

-order Astigmatism for Object at Infinity………………102

-Order Astigmatism for Object and

Image at Equal Distances…………………………………………………110

MEMS Mirror Represented as an Even Polynomial with 2 nd

- and 4 th

-order Coefficients ......................................................................................... 114

5.

EXPERIMENTAL VERIFICATION ....................................................................... 116

Optical Test Layout................................................................................................... 116

Results ....................................................................................................................... 117

Process-1 4.000 mm x 5.657 mm 45 o

Incidence Angle Mirror ......................... 118

Process-1 4.000 mm x 4.015 mm 5 o

Incidence Angle Mirror ........................... 122

6.

DEMONSTRATION OF FOCUS CONTROL AND SPHERICAL

ABERRATION CORRECTION FOR MULTI-LAYER DISCS ............................. 124

MEMS Requirements for Focusing and Aberration Correction of

Multi-layer DVD and BDXL

TM

Discs ...................................................................... 126

Demonstration of Necessary Performance................................................................ 127

Results ....................................................................................................................... 129

Chapter Summary ..................................................................................................... 135

7.

AGILE SCANNING USING A MEMS FOCUS CONTROL MIRROR

IN A COMMERCIAL CONFOCAL MICROSCOPE ............................................. 137

Optical Design .......................................................................................................... 139

Optical Characterization ........................................................................................... 141

Imaging Results ........................................................................................................ 146

8.

DISCUSSION ........................................................................................................... 151

APPENDICES ................................................................................................................ 157

APPENDIX A: Verification of -function with Zemax Ray Tracing

Results for Stop at Mirror ..................................................................... 158

APPENDIX B: Verification that 2 nd

and 3 rd

Order Terms Accurately

Represent -function ............................................................................ 177

APPENDIX C: Zernike Definitions................................................................................ 181

APPENDIX D: Electrostatic Pressure versus Displacement .......................................... 184

REFERENCES CITED ................................................................................................... 188

vi

LIST OF TABLES

Table

1.

Page

Three SU-8 processes and their resulting intrinsic stress of the membrane. ....................................................................................................... 23

2.

Change in Zernike defocus and spherical aberration coefficients for a process-1 3.00 mm x 3.01 mm mirror. ..................................... 31

3.

Change in Zernike defocus and spherical aberration coefficients for a process-2 5 mm 0 o

mirror.. ....................................................... 31

4.

Corresponding natural focus at distance,

r

2

, and

F

-number,

F,

for a given deflection,



and incidence angle,

2

, for and

a

= 2 mm. ....................................................................................................... 53

5.

Comparison of 3 rd

-order, variable-stop-position terms with

Mahajan’s axially symmetric terms. ..................................................................... 69

6.

Constraints and values given to variables to compare 3 rd

order aberration terms with Mahajan’s Equation 6-65b. ...................................... 69

7.

Axially symmetric aberration coefficients for an object lying at infinity at an angle

from the axis of a paraboloidal mirror. ................................................................................................................... 70

8.

Peak-to-valley aberration values for the case

= 30

 m, atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

, and

a

= 2 mm,

. ......................................... 77

9.

Comparison of an elliptical and a circular mirror used at 5 o incidence angle with

r

1

= 200 mm and

a

= 2 mm. ................................................ 94

10.

Maximum recommended angle of incidence for given deflections to maintain

S

= 0.8 for

= 633 nm,

r

1

= ∞, and

a

= 2 mm. ............................................................................................................ 100

11.

Maximum recommended field angle of atan( for given deflections to maintain

S u

2

/

r

2

) with

v

2

= 0

= 0.8 for

= 633 nm,

r

1

= ∞, and

a

= 2 mm. ......................................................................................... 100

12.

Maximum recommended field angle atan(

v

2

/

r

2

) with

u

2

= 0 for given deflections to maintain

S

= 0.8 for

= 633 nm,

r

1

= ∞, and

a

= 2 mm. ......................................................................................... 101

vii

LIST OF TABLES – CONTINUED

Table Page

13.

Maximum recommended field angle along the diagonal

(

u

2

=

v

2

) for given deflections to maintain

= 633 nm,

r

S

= 0.8 for

1

= ∞ and

a

= 2 mm. ...................................................................... 101

14.

Constraints and values given to variables to observe 1

of

2 nd

-order astigmatism. ......................................................................................... 103

15.

Constraints and values given to variables to observe 1

of

2 nd

-order astigmatism. ......................................................................................... 105

16.

Constraints and values given to variables to observe 1

of

2 nd

-order astigmatism. ......................................................................................... 108

17.

Constraints and values given to variables to observe 1

of

2 nd

-order astigmatism. ......................................................................................... 111

18.

Optical disc format specifications. ...................................................................... 125

19.

Focus control and spherical aberration correction requirements for multi-layer DVDs and BDs. .................................................... 126

20.

Radially symmetric Zernike polynomials of concern. ........................................ 142

21.

These voltage configurations improved the axial PSFs

(Figure 80) when compared with equipotential voltages on the electrodes. ..................................................................................................... 145

22.

Constraints and values given to variables to compare Matlab code with Zemax. ................................................................................................ 162

23.

Constraints and values given to variables to compare Matlab code with Zemax. ................................................................................................ 164

24.

Constraints and values given to variables to compare Matlab code with Zemax. ................................................................................................ 166

25.

Constraints and values given to variables to compare Matlab code with Zemax. ................................................................................................ 168

viii

LIST OF TABLES – CONTINUED

Table Page

26.

Constraints and values given to variables to compare Matlab code with Zemax. ................................................................................................ 171

27.

Constraints and values given to variables to compare Matlab code with Zemax. ................................................................................................ 173

28.

Constraints and values given to variables to compare Matlab code with Zemax. ................................................................................................ 175

29.

Constraints and values given to variables to compare Matlab code with Zemax.. ............................................................................................... 178

30.

Constraints and values given to variables to compare Matlab code with Zemax.. ............................................................................................... 180

31.

List of Zernike term definitions. ......................................................................... 183

ix

LIST OF FIGURES

Figure

1.

(left) Traditionally, translation of a glass lens leads to a change in

Page image location for a single lens. (right) Active elements differ in that they remain stationary while changing their optical properties to change their optical power. ................................................................................. 2

2.

(a) Hexagonal electrode pattern. MEMS mirrors with 32 or more actuators provide pixelated control of aberrations. (b) Concentric rings electrode pattern allow for control over defocus and spherical aberration while minimizing control complexity and overall cost. ............................................................................................................. 4

3.

(a) Schematic of a MEMS mirror with a variable deflection

that has an inverse relationship with the focal length of the reflected rays. (b) Picture of a mounted 1 mm x 1.4 mm MEMS mirror with thumb for scale. ....................................................................................................... 6

4.

Disc-read optical layout with quadruple-layer BDXL

TM

disc layer specifications shown. Deflection,



of the MEMS mirror shifts the focus of the beam from the deepest layer, L0, to layer L3, which is 46.5

 m closer to the objective lens than L0 [70]. ................................... 6

5.

(a) A 1

 m thermal oxide layer is grown and 150 nm PSG is deposited. (b) 2.5

 m of SU-8 2002 is photo-patterned. (c) 60 Å chrome and 150 nm gold are evaporated as a reflective coating and to form the top electrodes, patterned using a liftoff process;

60 Å chrome and 200 nm gold are evaporated onto the backside for the silicon counter electrode. (d) An air gap is created by wet etching of the oxides and silicon through the small vias in the SU-

8 and gold layers [1]. .............................................................................................. 8

6.

Cross section of the process-3 SU-8 2002 mirrors, which are dryetch released in XeF

2

. 2.5

 m of SU-8 2002 is photo-patterned;

60 Å chrome and 150 nm gold are evaporated as a reflective coating and to form the top electrodes, patterned using a liftoff process; 60

Å

chrome and 200 nm gold are evaporated onto the backside for the silicon counter electrode; an air gap is created by dry etching the silicon through the small vias [1]. .................................................. 9

x

LIST OF FIGURES – CONTINUED

Figure

7.

Page

Group 7 elements 4-6 USAF target images with the corresponding focus adjustment

f

[1]. Element 6 has 228 lp/mm. Electrode 1 is the inner-most electrode and electrode 3 actuates the perimeter.

All three electrodes have equal change in radius length. ...................................... 11

8.

Acquisition modes with a confocal microscope. (a) Traditional method. Several

x-y

planes are acquired and stitched together to form a three-dimensional

x-y-z

stack. (b) Imaging of an oblique plane. (c) Imaging of a specific z-trajectory to follow relevant biological structures for time-critical imaging and/or to reduce photobleaching of the sample. .............................................................................. 12

9.

Three

en face

sections and an oblique section of

Muscodor albus

fungus taken using MEMS focus control [77]. ..................................................... 12

10.

(a) Picture of two 2 mm x 2.82 mm elliptical-boundary mirrors mounted and wire-bonded on a TO5 header. (b) Optical interference micrograph of a 1 mm x 1.4 mm elliptical boundary mirror. (c) Surface profile of a 1 mm x 1.4 mm ellipticalboundary mirror. ................................................................................................... 19

11.

A 200x scanning electron microscope image of a 1 mm fixedperimeter, circular-boundary mirror after xenon diflouride etching with the top membrane removed. The etch vias were above the centers of the numerous dark circles. The lighter gray areas are higher silicon peaks where the etch fronts merged [1]. ........................................ 20

12.

(left) Cross section of the unreleased mirror. 2.5

 m of SU-8 2002 is photo-patterned on an SOI wafer. 60 Å chrome (Cr) and 160 nm gold (Au) are evaporated as a reflective coating and to serve as the top electrodes. 200 nm Au is evaporated onto the bottom side for the silicon counter-electrode. (right) 4

 m vias in the SU-

8 and 5

 m vias in the gold layer allow for dry-etching of the silicon with xenon difluoride to create an air gap. ................................................ 24

13.

(left) Tilted interferogram showing the flatness of the process-1

3 mm x 3.011 mm mirror. (right) Top view with labels on the four electrodes. ............................................................................................................. 24

xi

LIST OF FIGURES – CONTINUED

Figure Page

14.

A process-3 4 mm X 5.66 mm mirror with 466 V and a maximum deflection of 22

 m (

= 850 nm). ........................................................................ 28

15.

Voltage versus deflection for several mirrors. The perimeters of the mirrors experienced lateral over-etching of 100-200

 m, which makes interpretation of their voltage profiles complex. One can observe that smaller mirrors, larger air gaps, and higher intrinsic stress mirrors require more voltage for a given displacement. Snapdown into a 50

 m air gap limits the maximum displacement to 22

 m. ........................................................................ 28

16.

(left) Normalized magnitude plot of a process-2 5 mm 0 o

mirror with 3-dB response shown. (right) Phase response of a process-2

5 mm 0 o

mirror. ..................................................................................................... 29

17.

(left) Surface profile of a process-1 4 mm 45 o

mirror scaled to a circular perimeter as measured by phaseshift interferometry.

(right) a photograph of the mirror tilted in the interferometer for a flat (no voltage applied) process-1 4 mm 45 o

(4.000 mm x

5.657 mm) mirror. Lateral under-cutting extends the membrane beyond the optical surface to 4.340 mm x 5.997 mm. This relatively large MEMS mirror exhibits excellent optical quality with 147 nm peak-to-valley and 22 nm RMS. ...................................................... 32

18.

(left) Surface profile of a process-1 4 mm 5 o

mirror scaled to a circular perimeter as measured by phaseshift interferometry.

(right) A photograph of the mirror tilted in the interferometer for a flat (no voltage applied) process-1 4 mm 5 o

mirror. The MEMS mirror exhibits excellent optical quality with 113 nm peak-tovalley and 11 nm RMS. ........................................................................................ 32

19.

A flat (no voltage applied) process-1 4 mm 45 o

mirror.. ...................................... 34

20.

Wavefronts and Zernike coefficients for a process-1 4 mm 5 o mirror used in Chapter 5. All Zernike modes above 6 are 10x. The surface heights are reconstructions without defocus, tip, tilt, and piston (modes 1, 2, 3, and 5), but include all other terms. With an equipotential voltage of 0 V, the mirror is flat. An equipotential voltage of 200 V yields 5

 m mirror sag. The mirror sag is

10.9

 m at 247 V. ................................................................................................. 35

xii

LIST OF FIGURES – CONTINUED

Figure

21.

Comparison of Zernike terms and wavefronts for a process-1

4 mm 45 o

mirror with equipotential (233 V) and varied voltages to minimize spherical aberration terms (E1 = 234.5 V,

E2 = 198 V, E3 = 255 V, and E4 = 285 V). For both voltage profiles the center of the mirror is displaced 10

 m. (a) Zernike terms as generated from a phase-shift interferometer with normal

Page incident light rays on the mirror. Terms above Z5 are magnified by 10x. Primary and secondary spherical aberration decrease for the varied voltage case, where the varied voltages reduce the primary spherical aberration by approximately 300 nm. (b)

Equipotential wavefront. (c) The wavefront with spherical aberration minimized. ........................................................................................... 36

22.

Comparison of Zernike terms and wavefronts for a process-1

4 mm 45 o

mirror with equipotential (184.5 V) and varied voltages to minimize spherical aberration terms (E1 = 195 V, E2 = 173 V,

E3 = 194 V, and E4 = 206 V). For both voltage profiles the center of the mirror is displaced 5

 m. (a) Zernike terms as generated from a phase-shift interferometer with normal incident light rays on the mirror. Terms above Z5 are magnified by 10x. Primary and secondary spherical aberration decrease for the varied voltage case, where the varied voltages reduce the primary spherical aberration by approximately 700 nm. (b) Equipotential wavefront.

(c) The wavefront with spherical aberration minimized. ...................................... 37

23.

Comparison of Zernike terms and wavefronts for a process-1

4 mm 5 o

mirror with equipotential (200 V) and varied voltages to minimize spherical aberration terms (E1 = 200 V, E2 = 185 V,

E3 = 205 V, and E4 = 225 V). For both voltage profiles, the center of the mirror is displaced 5

 m. (a) Zernike terms as generated from a phase-shift interferometer with normal incident light rays on the mirror. Terms above Z5 are magnified by 10x.

Primary and secondary spherical aberration decrease for the varied voltage case, where the varied voltages reduce the primary spherical aberration by approximately 500 nm. (b) Equipotential wavefront. (c) The wavefront with spherical aberration minimized. ............................................................................................................ 38

xiii

LIST OF FIGURES – CONTINUED

Figure

24.

(a) Global coordinate system with origin

O

at the center of the mirror plane. The object plane contains point

P

10

and lies perpendicular to the object-space, base-ray . The image-

Page space, base-ray lies perpendicular to the image plane where the image plane also contains point

P

20

. (b) Local object (

u

1

, v

1

) coordinate plane perpendicular to . ............................................................. 40

25.

Local exit pupil (

x, y

) coordinate plane for determination of aberration coefficients. The exit pupil plane contains the origin

O

of the mirror plane and is normal to the image-space, base-ray

. ................................................................................................................... 41

26.

(a) The law of reflection restricts the entrance and exit angles of the base rays

r

1

and

r

2

with respect to the varifocal mirror to be equal in magnitude and opposite in sign. (b) Detailed drawing depicting the chief ray and ray of interest through point

Q

for the

-function. ............................................................................................................ 41

27.

A change in position of stop by distance

p

changes the reference chief ray from

CR

to

CR p

for calculation of the aberration function. ................................................................................................................ 47

28.

The actual ray from point

Q

to

P

2

crosses the exit pupil at (

x

’,

y

’), but our analysis provides the ray location as (

x

,

y

). .............................................. 48

29.

A typical elliptical-boundary mirror, where the perimeter of surface

Z

(

X, Y

) must be at a constant displacement value

. ................................ 49

30.

Mapping an ellipse with semi-minor axis

a

and semi-major axis

b

in coordinates onto a normalized circle in coordinates. ........................................................................................................... 50

31.

For the two astigmatic focal lines to coincide in

Z

,

b

=

a

/cos(-

2

)

=

a

/cos(

1

) must be maintained. Alternatively, the diameter of the major-axis, 2

b,

must project as the same diameter of the minoraxis, 2

a,

at a given angle of incidence.. ................................................................ 52

xiv

LIST OF FIGURES – CONTINUED

Figure Page

32.

For infinite-conjugate imaging with zero field, an off-axis paraboloidal (OAP) mirror introduces no aberration to the system.

A dynamic MEMS mirror with plane symmetry introduces a small amount of 2 nd

-order, field-independent aberration that presents similar to coma. The wavefronts shown have

r

2

= 150 mm or

= 9.4

 m for an

2

= 45 o

mirror. ............................................................ 57

33.

Second-order ray aberrations for a mirror with minor axis radius

a

= 2 mm and case . (a) Astigmatism with linear dependence on atan(

u

2

/

r

2

) = atan(

v

2

/

r

2 field variables, where

) = 1 o

. Given the linear nature of the field dependence, one can multiply the results by any value of

u

2

or

v

2 to determine aberrations for larger fields. ............................................................. 60

34.

The 3 rd

-order, peak-to-valley aberrations for various deflections and incidence angles of an

a

= 2 mm mirror and

Although atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

.

for all plots, the vertical axis labels show dependence of the aberration terms on field angles for ease of changing field angles. Plots a, b, c, d, e, f, g, h, i, j, and k are for coefficients 𝑑 and 𝑑

14

1

, 𝑑

2

, 𝑑

3, 𝑑

4

, 𝑑

5, 𝑑

6

, 𝑑

8

, 𝑑

9

, 𝑑

11

, 𝑑

12

,

. .................................................................................................................. 76

35.

The 3 rd

-order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror with . These two aberration terms, dependence on

 𝑑9, 𝑟1 → ∞

, and 𝑑14, 𝑟1 → ∞

, have no

2

. The field angle along the diagonal means that

u

2

=

v

2

. For instance, if atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 3.54

o

, then field angle along the diagonal is 5 o

. .............................................................................. 79

36.

The 3 rd

-order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 0 o

incidence angle with

. The term 𝑑6, 𝑟1 → ∞

is zero at normal incidence. The field angle in the

u

2 direction is atan(

u

2

/

r

2

). .......................................................... 80

37.

The 3 rd

-order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 5 o

incidence angle with

. The field angle in the

u

2 direction is atan(

u

2

/

r

2

). The field angle in the

v

2 direction is atan(

v

2

/

r

2

). .......................................................... 81

xv

LIST OF FIGURES – CONTINUED

Figure

38.

The 3 rd

order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 10 o

incidence angle with

Page

. The field angle in the

u

2 direction is atan(

u

2

/

r

2

). The field angle in the

v

2 direction is atan(

v

2

/

r

2

). .......................................................... 82

39.

The 3 rd

order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 45 o

incidence angle with

. The field angle in the

u

2 direction is atan(

u

2

/

r

2

). The field angle in the

v

2 direction is atan(

v

2

/

r

2

). .......................................................... 83

40.

Wavefront RMS for

2

= 0 o

, 5 o

, 10 o

, and 45 o

shown in rows. The left column shows no-field imaging with

u

2

=

v

2

= 0 and the right column has 5 o

of field along the diagonal in the image plane. All graphs show the mirror with a deflection ranging from 0

 m to

40

 m. ................................................................................................................... 86

41.

Wavefront RMS for

= 0

 m to 40

 m for 5 o

incidence angle.

The top plot shows a field of 3 o

along the diagonal. The bottom two plots show the aberrations with 3 o

field in the individual field coordinates. ........................................................................................................... 88

42.

Wavefront RMS for



= 0

 m to 40

 m for 45 o

incidence angle.

The top plot ........................................................................................................... 89

43.

Wavefront RMS for MEMS mirrors with different radii. All graphs show results for

2

= 0 o

with a deflection range of 0

 m to

40

 m. Mirrors with larger radii have less optical power for the same deflection as smaller radii mirrors. .............................................................. 90

44.

Wavefront RMS for MEMS mirrors with different radii. All graphs show results for

2

= 45 o

with a deflection range of 0

 m to 40

 m. ............................................................................................................... 91

45.

Wavefront RMS for

2

= 0 o

, 5 o

, 10 o

, and 45 o

shown in rows. The left column shows no-field imaging with

u

2

=

v

2

=0 and the right column has 5 o

of field along the diagonal in the image plane. All graphs show the mirror with a deflection ranging from ....................................... 92

xvi

LIST OF FIGURES – CONTINUED

Figure Page

46.

The 2 nd

-order terms, 3 rd

-order terms, and total peak-to-valley coma and astigmatism decrease with increasing

F

for an

2

= 5 o and

a

= 2 mm mirror with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

at

r

1

= ∞. ....................... 96

47.

The 2 coma and astigmatism decrease with increasing and

a

nd

-order terms, 3 rd

-order terms, and total peak-to-valley

= 2 mm mirror with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

F

for an

2

= 45 o

) = 1 o

at

r

1

= ∞. ....................... 97

48.

Peak-to-valley coma and astigmatism versus angle of incidence,

2

, for an

F

= 50 system with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

and

a

= 2 mm for

. ................................................................................. 98

49.

Field angle along the diagonal (where image plane variables have the opposite sign as

r

2

) in the image plane versus incidence angle for which RMS of either 2 nd

-order or 3 rd

-order aberrations dominate for

r

1

= ∞. The field angle at which 3 rd have a larger RMS than 2 nd

-order aberrations

-order aberrations remains the same for all deflections,



of the mirror. The total RMS for a mirror at

10

 m deflection is also shown. ............................................................................ 99

50.

Zemax optical path difference fans for four different image locations (

u

2

,v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial

(

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is 110 mm. ............................................................................ 103

51.

Zemax wavefront surfaces without tilt for four different image locations (

u

2

,v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 7.071 mm

( elliptical-boundary mirror represented as an even polynomial

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is 110 mm. ............................................................................ 104

52.

Zemax through-focus spot diagrams for four different image locations (

u

2

,v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial

(

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is 110 mm. ............................................................................ 104

xvii

LIST OF FIGURES – CONTINUED

Figure Page

53.

Zemax optical path difference fans for a 5 mm circular-boundary mirror represented

(

Z

= .00224

X

2

+.00224

Y

2 as

) with

 an

= 1

 even m, where

r

1 polynomial and

r

2

are infinity and 112 mm, respectively. (b) Wavefront surfaces. (c)

Through-focus spot diagrams at the four different image locations. ............................................................................................................. 106

54.

Zemax wavefront surfaces without tilt for a 5 mm circularboundary

(

Z

= .00224

X

mirror

2

+.00224

represented

Y

2 as an even

) with

= 1

 m. Variable

r

1 polynomial is infinity and

r

2

is 112 mm. ................................................................................................ 106

55.

Zemax through-focus spot diagrams for a 5 mm circular-boundary mirror represented

(

Z

= .00224

X

2

+.00224

Y

2 as

) with

 an

= 1

 even m. Variable polynomial

r

1 is infinity and

r

2

is 112 mm. ................................................................................................ 107

56.

Zemax optical path difference fans for four different image locations (

u

2

, v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 5.019 mm

( elliptical-boundary mirror represented as an even polynomial

Z

= .00226

X

2

+.00227

Y

2

) is analyzed here with

= 1

 m.

Variable

r

1 is infinity and

r

2

is 110 mm. ............................................................. 109

57.

Zemax wavefront surfaces without tilt for four different image locations (

u

2 due to 2 nd

, v

2

) to verify approximately 1 wave of astigmatism

-order aberration terms. A 5.000 mm x 5.019 mm

( elliptical-boundary mirror represented as an even polynomial

Z

= .00226

X

2

+.00227

Y

2

) is analyzed here with

= 1

 m.

Variable

r

1 is infinity and

r

2

is 110 mm. ............................................................. 109

58.

Zemax through-focus spot diagrams for four different image locations (

u

2

, v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 5.019 mm

( elliptical-boundary mirror represented as an even polynomial

Z

= .00226

X

2

+.00227

Y

2

) is analyzed here with

= 1

 m.

Variable

r

1 is infinity and

r

2

is 110 mm. ............................................................. 110

xviii

LIST OF FIGURES – CONTINUED

Figure Page

59.

Zemax optical path difference fans to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial (

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variables

r

1 and

r

2

are 221 mm. ......................................................... 112

60.

Zemax wavefront surfaces to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x

7.071 mm elliptical-boundary mirror represented as an even polynomial (

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variables

r

1 and

r

2

are 221 mm. ......................................................... 112

61.

Zemax through-focus spot diagrams to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial (

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variables

r

1 and

r

2

are 221 mm. ......................................................... 113

62.

A point source at

u

1

MEMS mirror at 45

=

v

1

= 0 in the object plane illuminates a o

incidence angle. The image plane has coordinates of

u

2

,

v

2

. The exit pupil plane (

x

,

y

coordinates) is normal to the image plane. .................................................................................. 116

63.

(a) A point source (



= 633 nm) illuminates the MEMS mirror from a distance,

r

1

. A 1x relay consisting of two

f

= 30 mm achromats images the MEMS mirror onto the wavefront sensor.

When the natural focus of the MEMS mirror equals

r

1

, the MEMS mirror collimates the green light rays as shown. The exit pupil plane is imaged onto the WFS. (b) The same optical set-up as shown in (a) with local coordinate systems shown. ............................................ 117

64.

Wavefronts generated from the entire -function, 2 nd

- and 3 rd

order aberration terms, and Zernike terms from the WFS for

r

1

=

284 mm. .............................................................................................................. 119

65.

Wavefronts generated from the entire -function, 2 nd

and 3 rd order aberration terms, and Zernike terms from the WFS for

r

1

=

141 mm. .............................................................................................................. 120

xix

LIST OF FIGURES – CONTINUED

Figure Page

66.

Wavefronts generated from the entire -function, 2 nd

- and 3 rd

order aberration terms, and Zernike terms from the WFS for

r

1

=

76 mm. ................................................................................................................ 121

67.

Wavefronts generated from the entire -function to demonstrate theoretical aberrations for no-field imaging for

r

1

= 284 mm,

r

1

=

141 mm, and

r

1

= 76 mm. ................................................................................... 121

68.

Wavefronts generated from the entire -function, 2 nd

- and 3 rd

order aberration terms, and Zernike terms from the WFS for

r

1

=

284 mm. Note: The shape of the MEMS mirror was not minimized to reduce spherical aberration, as was done for the 45 o incidence mirror results above. ........................................................................... 123

69.

Interferometer for observing aberrations at the DVD objective lens. ..................................................................................................................... 129

70.

(a) Interferogram with an optical flat in place of the MEMS mirror and cover glass with thickness of 620

 m. (b)

Interferogram with un-deflected MEMS mirror and 620

 m cover glass; both (a) and (b) show nearly flat fringes indicating the absence of spherical aberration. (c) Interferogram with 370

 m thick cover glass shows approximately 3.2

 m of spherical aberration in double-pass reflection, corresponding to 1.6

 m single-pass. (d) Interferogram with 200 V on all three electrodes, peak mirror deflection is 6.7

 m and round trip aberration is reduced by approximately 1 fringe. (e) Three-zone control minimized single-pass spherical aberration to less than 130 nm peak-to-valley. .................................................................................................... 131

71.

The 6 th

-order, even-polynomial fits ( ) to the measured displacements of the membrane with (a) 200 V equipotential, a center displacement of 6.7

 m, and

a

2

= 7.72,

a

4

=

-1.85 and

a

6

= 0.74 and (b) 189 V, 202 V, and 225 V on electrodes

1, 2, and 3, respectively, a center displacement of 7

 m, and

a

2

=

7.50,..................................................................................................................... 133

xx

LIST OF FIGURES – CONTINUED

Figure Page

72.

Spherical aberration introduced by the surface shape of the membrane including (a) plots of only and (b) the same data with balancing defocus added. .................................................................... 134

73.

Asymmetric aberration evident when defocus and tilt are removed, and spherical aberration is balanced along the vertical aspect of the pupil (the short axis of the membrane). ......................................... 134

74.

(left) Tilted interferogram showing the flatness (less than 90 nm peak-peak deviation) of the process-1 3-mm x 3.011-mm mirror.

(right) Top view with labels on the four electrodes. ........................................... 139

75.

Olympus Fluoview 300 with MEMS module shown in the green box. The MEMS module consists of an f = -166 mm lens to push the focal plane in the specimen out from the objective when the

MEMS mirror is flat............................................................................................ 140

76.

The natural focal plane of the Olympus water objective is defined as 0

 m. With the MEMS module (including the negative lens) inserted with an un-deflected MEMS mirror the focus lies at -

40

 m. An increase in voltage on the MEMS mirror increases its displacement until the focal plane behind the objective lens lies at

83

 m. ................................................................................................................. 141

77.

Three voltage profiles with corresponding surface shapes and

Zernike terms. The amplitude of each Zernike mode higher than mode 5 (defocus) is shown plotted 10x the measured amplitude for all three spectra. Mode 13 is primary spherical and mode 25 is secondary spherical. (a) A voltage of 350 V on the three inner electrodes and 400 V on the outermost electrode results in a parabolic shape (introducing almost no spherical aberration). (b)

Much higher voltage on the outer rings than the centermost electrode yields 230 nm of positive primary spherical aberration and -70 nm of secondary spherical aberration. (c) Having the greatest electrostatic pressure in the center of the membrane results in a more cone-like surface as evidenced by a smaller area of blue (greater than -3000 nm) in the surface profile. It produces

-180 nm primary and 75 nm secondary spherical aberration. ............................. 142

xxi

LIST OF FIGURES – CONTINUED

Figure Page

78.

Axial point spread function (



= 633 nm) of the microscope with no MEMS module inserted. ................................................................................ 143

79.

Axial point spread functions (



= 633 nm) of the microscope with the MEMS module inserted and equipotential voltages applied.

The MEMS mirror begins nearly flat for the PSF at an axial displacement of -25

 m. As the MEMS mirror increases power, the axial displacement increases in the positive direction. ................................. 144

80.

Altering voltages on the four electrodes improves the axial PSFs

(



= 633 nm) while maintaining similar focal planes as in Figure

79......................................................................................................................... 145

81.

Axial point spread functions (



= 488 nm) of the confocal microscope for three different voltage profiles. Defocus remained constant for the three measurements. All horizontal (axial displacement) and vertical (intensity) axes are the same scale. (a)

The point spread function has several large side lobes with the center electrode’s voltage being 160 V less than the outer two electrodes. (b) The side lobes shift to the left-hand side of the main lobe with 350 V on the centermost electrode and 300 V on the outer electrodes. (c) With 300 V on the inner two and 400 V on the outer two electrodes point spread function gains intensity on the central lobe and it narrows to approximately 5

 m FWHM. ................... 146

82.

Comparison of focus control by changing electrostatic pressure on a MEMS mirror at 5 o

incidence angle (top) versus translating the microscope while leaving the MEMS mirror flat (bottom). The image quality appears comparable for both mechanisms. .................................. 147

83.

Images of a Stage 28 chick embryo spinal cord, comparing focusing using z-stage translation to focusing with a MEMS mirror at 45 o

incidence angle. ............................................................................. 147

xxii

LIST OF FIGURES – CONTINUED

Figure Page

84.

(a) Oblique scan starting at

z

= -40

 m, increasing to

z

= 83

 m, and returning quickly to

z

= -40

 m to begin the next line at a new

y

-coordinate. The change in

z

-direction occurs over 9 ms. (b) An oblique scan occurring in 4.5 ms or at 32 o

angle in the sample. Different features present in the first 100

 m of the

x

coordinate than at a 16 o

angle oblique scan due to a different profile in

z

. .......................................................................................................... 148

85.

Comparative axial point spread functions (same scaling) and single

x-y

plane fluorescent images of a honey bee stinger, all taken at 543 nm. The voltages correspond with a focal point in the sample at 83

 m in

z

-direction. ..................................................................... 149

86.

Wide field (±1.7

o

x ±1.3

o

) images taken with a 0.4 NA 40x microscope with a 45 o

incidence angle mirror that should have less than a 0.8 Strehl ratio. .................................................................................. 154

87.

Zemax spot diagram of a 20 o

incidence angle 4 mm x 4.26 mm elliptical mirror with 5

 m deflection and an object at infinity with no field. ....................................................................................................... 159

88.

(left) For greatly divergent rays, the rays from the object miss the far edge of the MEMS mirror and never reach the exit pupil.

(right) The Hamiltonian analysis assumes the exit pupil is filled or that the MEMS aperture may be extended to ensure all rays pass through the exit pupil and reach the image (

r

2

= -13.944 mm for the case shown). .................................................................................................. 160

89.

OPD fans in Zemax for circular mirror with normal incidence, no field,

= 20

 m, and

r

1

= 200 mm. .................................................................... 162

90.

OPD fans from -function for circular mirror with normal incidence,

= 20

 m, no field, and

r

1

= 200 mm. .............................................. 163

91.

OPD fans in Zemax for 5 o

incidence mirror,



= 2

 m,

u

2

= 0.5 mm,

v

2

= 0 mm, and

r

1

= ∞. .................................................................. 164

92.

OPD fans from -function for 5 o

incidence mirror,

= 2

 m,

u

2

= 0.5 mm,

v

2

= 0 mm, and

r

1

= ∞. .................................................................. 165

xxiii

LIST OF FIGURES – CONTINUED

Figure Page

93.

Lens Data Editor and 3D layout in Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. ...................................... 166

94.

OPD fans in Zemax for 45 o

incidence mirror,



= 20

 m,

u

2

=

0 mm,

v

2

= 0 mm, and

r

1

= ∞. ............................................................................. 167

95.

OPD fans from -function for 45 o

incidence mirror,



= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. ...................................................................... 167

96.

Lens Data Editor and 3D Layout in Zemax for 45 mirror,

= 20

 m,

u

2

= 5 mm,

v

o

incidence

2

= 0 mm, and

r

1

= ∞. ....................................... 169

97.

OPD fans from Zemax for 45

u

2

= 5 mm,

v

2

= 0 mm, and

r

1 o

incidence mirror,



= 20

 m,

= ∞. ...................................................................... 169

98.

OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 5 mm,

v

2

= 0 mm, and

r

1

= ∞. ...................................................................... 170

99.

Lens Data Editor and 3D Layout in Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 5 mm, and

r

1

= ∞. ..................................... 171

100.

OPD fans from Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 5 mm, and

r

1

= ∞. .................................................................... 172

101.

OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 5 mm, and

r

1

= ∞. .................................................................... 172

102.

OPD fans from Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= 200 mm. ......................................................... 174

103.

OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= 200 mm. ......................................................... 174

104.

OPD fans from Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 5 mm,

v

2

= 5 mm, and

r

1

= 200 mm. ......................................................... 176

105.

OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 5 mm,

v

2

= 5 mm, and

r

1

= 200 mm. ......................................................... 176

xxiv

LIST OF FIGURES – CONTINUED

Figure

106.

(left) Comparison of total calculated -function, parabasal, 2 nd

- order aberration contribution, and 3 rd

- order aberration contribution for a parabolic mirror with 5 o

incidence angle,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. The optical path difference is shown for

x

in the pupil plane (tangential plane).

Page

(right) The optical path difference and Taylor series expansion of the -function are shown for

y

in the pupil plane (sagittal plane). ................... 179

107.

(left) Comparison of total calculated -function, parabasal, 2 nd

- order aberration contribution, and 3 rd

- order aberration contribution for a parabolic mirror with 45 o

incidence angle,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. The optical path difference is shown for

x

in the pupil plane (tangential plane).

(right) The optical path difference and Taylor series expansion of the -function are shown for

y

in the pupil plane (sagittal plane). ................... 180

108.

Electrostatic pressure schematic. ....................................................................... 185

xxv

ABSTRACT

Micro-electro-mechanical systems deformable-membrane mirrors provide a means of focus control and attendant spherical aberration correction for miniaturized imaging systems. The technology has greatly advanced in the last decade, thereby extending their focal range capabilities.

This dissertation describes a novel SU-8 2002 silicon-on-insulator wafer deformable mirror. A 4.000 mm x 5.657 mm mirror for 45 o

incident light rays achieves

22

 m stroke or 65 diopters, limited by snapdown. The mirrors show excellent optical quality while flat. Most have peak-to-valley difference of less than 150 nm and rootmean-square less than 25 nm. The process proves simple, only requiring a silicon-oninsulator wafer, SU-8 2002, and a metal layer. Xenon difluoride etches the silicon to release the mirrors. Greater than 90% of the devices survive fabrication and release.

While current literature includes several aberration analyses on static mirrors, analyses that incorporate the dynamic nature of these mirrors do not exist. Optical designers may have a choice between deformable mirrors and other types of varifocal mirrors or lenses. Furthermore, a dynamic mirror at an incidence angle other than normal may be desired due to space limitations or for higher throughput (normal incidence requires a beam splitter). This dissertation presents an analysis based on the characteristic function of the system. It provides 2 nd

and 3 rd

order aberration coefficients in terms of dynamic focus range and base ray incidence angle. These afford an understanding of the significance of different types of aberrations. Root-mean-square and Strehl calculations provide insight into overall imaging performance for various conditions. I present general guidelines for maximum incidence angle and field of fiew that provide near diffractionlimited performance. Experimental verification of the MEMS mirrors at 5 o

and 45 o incidence angles validates the analytical results.

A Blu-ray optical pick-up imaging demonstration shows the utility of these mirrors for focus control and spherical aberration correction. Imaging results of the first demonstration of a deformable mirror for dynamic agile focus control and spherical aberration correction in a commercial table-top confocal microscope are also shown.

1

CHAPTER 1

INTRODUCTION

History

The demand for small-form-factor imaging systems continues to push high quality optics to smaller sizes. Platforms necessitating miniature imaging systems include cell phone cameras, tablets, endoscopic microscopes and cameras, optical pick-ups, and picoprojectors. A key mechanism for these imaging systems is focus control. The traditional mechanism of translating glass lenses by motors and cams has proven difficult to

miniaturize. Figure 1 shows the simplest form of focus control, in which translation of a

single lens shifts the location of focus for the object. As an alternative to translating lenses, investigators have explored the use of single-element, variable-power optical

elements (Figure 1). Not only are these elements capable of defocus, but they often serve

as adaptive optics by correcting aberrations in optical systems. Investigators have demonstrated varifocal lenses in microscopy [1-9], ocular adaptive optics [10-14], cameras [15-22], astronomy [12, 23-25], optical interconnects [26-28], pulse compression or shaping [29-31], optical data storage [3, 4, 32-34], micromachining [4,

35, 36], and optical tweezers [3, 37, 38]. Active optical elements include deformablemembrane mirrors [16, 25, 39-48], liquid-filled lenses (including electrowetting) [49-58], and spatial light modulators [59-61]. Micro-electro-mechanical systems (MEMS) comprise the majority of such elements due to relative ease of fabrication and their small form factor.

2

Translation of lens

Object

Image

Object

Active optics

Image

Lens

Stationary element

Figure 1: (left) Traditionally, translation of a glass lens leads to a change in image location for a single lens. (right) Active elements differ in that they remain stationary while changing their optical properties to change their optical power.

The widespread use of cell phones makes them a target platform for many MEMS applications. According to Johnson [62], smartphones currently have up to 12 MEMS devices and we can expect to see an increase in future years. Research and Markets [63] expects new sensors for mobile applications to grow from $2.2B in 2012 to $6.4B by

2018 [63]. MEMS act as sensors or actuators for accelerometers, gyroscopes, microspeakers, pressure sensors, microdisplays, autofocusing, touchscreens, biochemical detection, and energy harvesting [63]. Optical MEMS investigators have begun to utilize the cell phone platform. Graham-Rowe [64] has cited three companies—Varioptic,

Philips of the Netherlands, and the Institute of Materials Research and Engineering— as competing for successful commercialization of a type of MEMS variable-optical-power element, the liquid lens, in cell phone cameras. Our research laboratory, in collaboration with Bridger Photonics, demonstrated the potential for electrostatic MEMS mirrors for focus control in small-form-factor cameras [15]. In a very different application, investigators have begun exploring different avenues to integrate microscopes into or for

3 use with cell phone cameras. More specifically in the area of optical MEMs, researchers have begun to explore the utility of a cell phone camera to serve as a platform for a microscope and spectroscope for identification of sickle cell anemia in developing countries [6]. As this technology progresses, optical MEMS may play a role in its miniaturization. In our highly electronically connected world, mobility guides current technological trends and, as such, pushes miniaturization of the components.

Performance metrics for variable-power elements include speed, maximum defocus capability, possibility for and quality of aberration compensation, and stability.

The availability of several different types of variable-power elements, plenoptic cameras

(typically lower resolution capabilities), and digital processing capabilities means the designer of an imaging system must decide on the best technology for their given application. An understanding of the imaging performance of MEMS mirrors for focus control under different conditions should aid the designer in making an optimal choice between the available technologies. Furthermore, for a given optical system, an understanding of the inherent aberrations of these mirrors should help guide future mirror structures and electrode patterns to take best advantage of their unique capabilities.

Deformable-Membrane MEMS Mirrors

Two main categories of deformable MEMS mirrors exist. The majority of MEMS mirrors on the market and investigated in academia primarily perform aberration correction with a small amount of defocus capability. The term adaptive means these mirrors can dynamically adapt their aberration balance by fine shape control with the use

of several (typically 35 or more) pixelated actuators. Figure 2a shows a common

4 hexagonal grid pattern for the electrodes; designers also frequently use a rectangular grid or segmented concentric circles. Of these pixelated adaptive mirrors, some have large stroke and perform significant defocus in addition to aberration correction. Our lab group

has focused on a unique approach: a small number of concentric electrodes (Figure 2b)

correct spherical aberration with the intention of minimizing complexity and overall cost.

For no-field imaging, such as in confocal microscopy, performing defocus while concurrently managing attendant spherical aberration may be sufficient for satisfactory image quality. This dissertation will concentrate primarily on deformable mirrors that are capable of large stroke or, equivalently, significant defocus.

(a) (b)

Figure 2: (a) Hexagonal electrode pattern. MEMS mirrors with 32 or more actuators provide pixelated control of aberrations. (b) Concentric rings electrode pattern allow for control over defocus and spherical aberration while minimizing control complexity and overall cost.

Deformable MEMS mirrors are variable-power, reflective optical elements that

have several compelling properties. Figure 3a shows a mirror with displacement,



that is inversely proportional to its paraxial focal length. MEMS mirrors tend to be less than

1 inch in diameter. The mirrors presented in this dissertation range from 750

 m to 5 mm

in nominal diameter (Figure 3b shows scale) to aid in miniaturization of imaging systems.

5

Figure 4 illustrates the utility of a MEMS mirror at 45

o

incidence angle for no-field electronic focus control in a quadlayer optical disk. With the membrane mirror flat, it exhibits no optical power and the Blu-ray objective lens focuses rays on the deepest layer, L0. With a deflection,



the rays focus at a shallower depth, up to layer L3 in the disk. These MEMS mirrors are not limited to no-field imaging for optical disk pick-ups and confocal microscope type of applications, but have also been demonstrated for finitefield-of-view imaging applications, such as small-form-factor cameras [15]. The relatively large wavefront sag possible with electrostatically actuated (or other means of actuation such as piezoelectric, thermal or magnetic) deformable mirrors provides a useful optical power range, typically exhibiting a focal range from infinity with a flat mirror to less than 100 mm for mirrors less than 4 mm in diameter. In the case of electrostatically actuated mirrors, the number and arrangement of electrodes proves flexible, allowing for adaptive surface shapes and a simple interface for control. (For instance, a 3 mm x 4.24 mm elliptical mirror discussed in Chapter 6 adjusted both primary (3 rd

order) and secondary (5 th

order) spherical aberration with only three concentric electrodes.) Due to low mass of the thin membrane, these elements can be quite fast (10s of kHz) in comparison with other similar technologies. Furthermore, reflective elements possess the advantage of being achromatic. Not surprisingly, several companies have commercialized deformable-membrane mirrors [65-69].

6

Object

Image

MEMS

Mirror

(a) (b)

Figure 3: (a) Schematic of a MEMS mirror with a variable deflection

that has an inverse relationship with the focal length of the reflected rays. (b) Picture of a mounted

1 mm x 1.4 mm MEMS mirror with thumb for scale.

L3

L2

L1

L0

MEMs

Mirror

Objective

Lens

15.5

 m

19.5

 m

11.5

 m

53.5

 m

Figure 4: Disc-read optical layout with quadruple-layer BDXL

TM

disc layer specifications shown. Deflection,



of the MEMS mirror shifts the focus of the beam from the deepest layer, L0, to layer L3, which is 46.5

 m closer to the objective lens than L0 [70].

7

Previous MEMS Mirror Development

The first published single-wafer, wet-released polymer (SU-8 2002) largedeflection mirrors minimized cost and fabrication complexity when compared with other common methods of MEMS mirror fabrication [1]. A wet-release process has the potential for release of several wafers at one time versus a more expensive dry-release process that is often limited to a single wafer. SU-8 2002 also has the advantage of being photo-patternable, which only requires photolithography equipment for processing in comparison with silicon nitride and other membrane materials that require additional vapor deposition equipment. Exploration of the photosensitive polymer’s process parameters minimized residual stress, significantly improving their overall focal range [1,

71]. A dry-etch process results in greater gap depth without damage to the SU-8 film and prevents stiction of larger dimension membranes, while also simplifying the process, eliminating the need for two oxide layers and their associated masks [1].

For the wet-release processes, 1

 m thermal oxide is grown on a <100> silicon

(Si) substrate to increase electrical isolation (Figure 5). Sacrificial oxide pads extending

to the perimeters of the mirrors are created by depositing 150 nm LPCVD 8%

phosphosilicate glass (PSG) (Figure 5a). The PSG allows for quick lateral etching by

BOE 6:1 to expose fully the thermal oxide for release of the membranes. A 2.5-

 m-thick

SU-8 2002 membrane is then spin coated and patterned with etch vias (Figure 5b). To

provide a reflective coating and top electrodes, 60 Å chrome and 150 nm gold are evaporated over 3-

 m-thick NR-1500PY Futurrex negative resist using a metal liftoff

procedure (Figure 5c). The patterned Futurrex consists of the electrode pattern as well as

8 plugs to protect each via and cutout to prevent metal from evaporating into the SU-8 openings. Finally, after protecting the front side with a thick layer of S1813, we evaporate a backside layer of 60 Å chrome and 200 nm gold. The metal coating improves electrical contact with the silicon counter electrode. During release, the oxides and silicon are wet etched through the 3-

 m-square vias by BOE 6:1 and TMAH, respectively

(Figure 5d).

The dry-release devices have the SU-8 2002 film directly on a <100> silicon

substrate (Figure 6). Because the release etch is isotropic, no PSG layer is required for

lateral etching so that layer is eliminated, and for simplicity we eliminated the thermal oxide film. The metallization steps are the same as for the wet-release processes. Etching of silicon in xenon difluoride (XeF

2

) in a XACTIX etcher releases the membrane.

(a) (b)

(c)

Silicon Oxide

(d)

PSG

SU-8 2002 Cr/Au Air Gap

Figure 5: (a) A 1

 m thermal oxide layer is grown and 150 nm PSG is deposited. (b)

2.5

 m of SU-8 2002 is photo-patterned. (c) 60 Å chrome and 150 nm gold are evaporated as a reflective coating and to form the top electrodes, patterned using a liftoff process; 60

Å

chrome and 200 nm gold are evaporated onto the backside for the silicon counter electrode. (d) An air gap is created by wet etching of the oxides and silicon through the small vias in the SU-8 and gold layers [1].

9

10

m

Silicon SU-8 2002 Cr/Au

Figure 6: Cross section of the process-3 SU-8 2002 mirrors, which are dry-etch released in XeF

2

. 2.5

 m of SU-8 2002 is photo-patterned; 60 Å chrome and 150 nm gold are evaporated as a reflective coating and to form the top electrodes, patterned using a liftoff process; 60

Å

chrome and 200 nm gold are evaporated onto the backside for the silicon counter electrode; an air gap is created by dry etching the silicon through the small vias

[1].

Further exploration of the tension and stress of SU-8 2002 under different processing parameters by Oliver et al. [71] suggests more pliable membranes are possible, indicating potential for greater stroke/optical power range. Furthermore, a capacitive control system, sensed directly at the electrode of the device, increased the maximum displacement of one of our mirrors by 150% when comparing performance of closed-loop versus open-loop control [72, 73]. Until recently, 10

 m stroke seemed the most typical displacement achieved by similarly sized mirrors with relatively simple fabrication schemes and materials. Wang et al. [17, 74] and Tsai et al. [75] have also been pushing the displacement limits of similar nominal-sized mirrors to improve performance. Wang et al. use a two-wafer process with an amorphous fluoropolymer,

CYTOP, membrane to achieve 27

 m deflection with a 4.5 mm diameter mirror at 160 V.

Hseih et al. use a two-wafer process with a polyimide membrane to achieve 12

 m displacement with a 3 mm diameter mirror (20 diopters). Both investigators use reasonably complicated fabrication schemes and require several etchants for release.

Greater deflection than the 15-20

 m we currently see for 3-5 mm devices proves advantageous for meeting focus control requirements of current applications. Realizing a

10 large focus range offers little benefit, however, if self-induced aberrations severely degrade image quality. These deformable-membrane mirrors naturally deflect parabolically for small displacements [76]. For infinite-conjugate, no-field imaging, a paraboloidal mirror shape introduces no aberrations. With greater displacements, however, the electrostatic pressure is non-uniform due to the varying air gap size in the capacitor. As a result, the mirror shape deviates from parabolic, leading to aberrations.

Furthermore, greater focus range may push the limits of other optics, such as objective lenses, in the system beyond their optimal design configuration.

Evaluation of Imaging

Performance of MEMS Mirrors

Compensation of aberrations arising from other optics in the system and selfinduced aberrations requires precise control of the optical surface shape. We demonstrated this in a simple 0.4 numerical aperture (NA) 40x microscope [1]. A 3 mm x

4.24 mm MEMS mirror with three concentric electrodes provided focus control at 45 o

incidence angle for the demonstration. The images in Figure 7 are from Group 7,

elements 4-6 of the 1951 USAF target. The smallest bars are 228 lp/mm. With no voltage

applied to the actuation electrodes, we observe a clear image (Figure 7a). With all three

electrodes connected in parallel with 196 V applied, the best focus is found at

f

= 137 μm (Figure 7b). The image contrast here is degraded, indicating aberration in

the system. Applying varying voltages on the three concentric electrodes eliminates

200 nm balanced spherical aberration and results in a pure defocus (parabolic) shape of

11

the MEMS mirror. These control voltages produce the image in Figure 7c, improving

image contrast.

(a) (b) (c)

0 V Electrode 1

0 V Electrode 2

196 V Electrode 1

196 V Electrode 2

178 V Electrode 1

197 V Electrode 2

0 V Electrode 3 196 V Electrode 3 230 V Electrode 3

Figure 7: Group 7 elements 4-6 USAF target images with the corresponding focus adjustment

f

[1]. Element 6 has 228 lp/mm. Electrode 1 is the inner-most electrode and electrode 3 actuates the perimeter. All three electrodes have equal change in radius length.

Chapter 6 contains a demonstration of the feasibility of deformable-membrane

Flat optical mirror Flat optical mirror mirrors for focus control and spherical aberration compensation in a Blu-ray disc pick-up.

There we show correction of wavefront aberrations of primary and secondary spherical aberration with peak magnitude in excess of 1.3

 m while defocusing the wavefront by

10

m by use of only three electrodes (Figure 70, Figure 71, Figure 72) [70]. We also

show the inability to correct residual asymmetric aberration with concentric electrodes

(Figure 73).

These studies examine the current capabilities of MEMS deformable mirrors to compensate spherical aberration while also exhibiting defocus. In addition to the Blu-ray disc application, Chapter 7 demonstrates the utility of these mirrors in a commercial,

table-top confocal microscope. Figure 8a shows the traditional data acquisition method

for confocal microscopy. The user acquires a 3D

x-y-z

stack for later data processing.

12

Figure 8b and Figure 9 show the capability of a MEMS mirror in a commercial confocal

microscope to perform oblique plane imaging by synchronization of the MEMS mirror with the

x

- or

y

-axis scanning mirror. Fast and agile focus control (Figure 8c) could

advance time-critical imaging of biological phenomena, as well as reduce photobleaching during longitudinal studies.. Chapter 7 not only investigates the effect of surface shape of the MEMS mirror on spherical aberration, but the effect the mirror shape has on the axial point-spread-function (PSF) of the entire system in different focal planes of a water objective lens.

(a) (b) (c)

Figure 8: Acquisition modes with a confocal microscope. (a) Traditional method. Several

x-y

planes are acquired and stitched together to form a three-dimensional

x-y-z

stack. (b)

Imaging of an oblique plane. (c) Imaging of a specific z-trajectory to follow relevant biological structures for time-critical imaging and/or to reduce photobleaching of the sample.

Figure 9: Three

en face

sections and an oblique section of

Muscodor albus

fungus taken using MEMS focus control [77].

13

Motivation for Aberration Analysis

Varifocal elements, whether refractive or reflective, are a relatively new technology. With regard to deformable-membrane mirrors, I previously discussed significant improvement in maximum displacements of these MEMS mirrors in the last few years due to more pliable membranes and closed-loop control techniques. Greater deflection leads to more significant inherent aberrations. The increasing use of these elements warrants a thorough understanding of how they perform in terms of aberrations.

The current approach to understanding aberrations introduced into an optical system by a

MEMS mirror involves expressing the aberrations of a built device in terms of Zernike polynomials [78-80]. This provides some insight to the designer as to what kind of performance one might expect of the mirror in an imaging system, but does not capture the overall limitations of the mirror in terms of field of view. It also fails to predict what kind of performance one might expect as maximum displacement continues to increase.

Effort also has been made to identify requirements for imaging applications with regard to focus control [81] and aberrations, such as spherical [81], and due to the inhomogeneous structure of biological specimens [3, 61, 82]. Again, these papers do not sufficiently address the inherent capabilities of deformable MEMS mirrors for meeting these requirements.

Sometimes non-circular mirrors with non-normal incident light are desirable to fit optical systems into tight space constraints. Moreover, non-normal incidence mirrors mitigate the need for a beam splitter. A common static off-axis mirror for ideal infiniteconjugate imaging at non-normal angles of incidence is an off-axis paraboloid. Himmer

14 et al. [105, 106] introduced the concept of deformable-membrane mirrors with elliptical boundaries to act as dynamic off-axis mirrors. These studies compared experimental polynomial curve fits of mirrors actuated up to 2.5

 m deflection with the ideal infiniteconjugate condition of an off-axis paraboloid (of equivalent paraxial focal length) to assess peak-to-peak wavefront aberration of the varifocal mirrors [105]. They compared the analytical equation for an elliptical-boundary paraboloidal mirror used off-axis with the equation for an off-axis paraboloidal mirror to calculate theoretical aberrations, but did not separate out the different types of aberrations (e.g., coma, astigmatism) that arise nor assess their impact for specific conditions.

An analytical approach to analyzing mirror aberrations is useful for understanding the inherent limitations of these mirrors over a broad focal range. Specific types of aberration trends and relationships to particular variables may be more apparent with an analytical result than only using ray-tracing software for analysis. Even an aspherical lens designed for one configuration may perform less well than a spherical lens when observing performance over a range of object and image locations. In addition, an analytical analysis should guide the pattern and number of electrodes for future mirrors, where every electrode increases complexity and cost to control. This analysis should benefit the future design of such mirrors and provide guidance when choosing these types of variable-focus mirrors for specific applications. These mirrors also perform reasonably well in a wide-field microscope [2, 40] and a camera [15]. These applications require a full-field aberration analysis to describe their behavior adequately.

15

Relevant Analyses of Aberrations for Varifocal Mirrors

An analysis based on the characteristic function of a single MEMS mirror system should provide the designer with the inherent limitations and capabilities of a dynamic varifocal mirror. Several analyses of aberrations for static reflectors exist [83-94]. Some are specific to both normal incidence [93, 95] and off-axis [91, 92, 96-103] paraboloidal or ellipsoidal mirrors. Hamiltonian analyses of static mirrors have been done previously for various shapes (spherical, conical) of mirrors out to 2 nd

order or 3 rd

order in terms of pupil- and image-coordinates [89, 104]. However, these analyses do not capture the dynamic nature of varifocal elements. More specifically, the physical structure of varifocal MEMS mirrors is not adequately represented by the variables in the aberration coefficients. Furthermore, much of the work on paraboloidal mirrors assumes infiniteconjugate imaging. While ellipsoidal mirrors are designed for finite object and image locations, the analysis by Murphy [103] neglects field considerations.

Contribution of This Work

This dissertation introduces the first in-depth analysis of off-axis aberrations for elliptical-boundary, varifocal mirrors under infinite- and finite-conjugate imaging conditions. One may note that a circle represents a special case of an ellipse; therefore, the analysis is valid for circular-boundary mirrors. The more common axially symmetric analyses may not be used for plane-symmetric systems, such as these mirrors, so a

Hamiltonian optics approach is used. This paper includes an aberration analysis that is relevant to current deformable-membrane mirror technology by capturing their

16 performance over a range of optical power (i.e., for different peak deflections of the membrane) and over a range of incidence angles (0-90 o

). By providing aberration coefficients in terms of incidence angle, the analysis can be extended to mirrors that incorporate tip/tilt and dynamic focus control [107]. I provide general guidelines for maintaining a 0.8 Strehl ratio in terms of field and incidence angle for these mirrors with parabolic deflection shape. The aberration analysis provides 2 nd

- and 3 rd

- order rayaberration coefficients for a MEMS mirror represented as both 2 nd

- and 4 th

- order even polynomials. A comprehensive treatment of the aberrations of varifocal mirrors should allow a designer to predict performance of a deformable mirror in a particular dynamic optical system. Knowledge of the inherent aberrations of the MEMS mirror by itself may also help guide system design. Additionally, the individual aberration terms are dependent on specific MEMS mirror shapes in the exit pupil. This knowledge may guide future design of electrode shapes and/or design of overall mirror structure.

The contribution of this work includes a novel design for a large-deflection, single-SOI-wafer, deformable-membrane mirror. We demonstrate a maximum center deflection of 22

 m. The mirrors exhibit excellent optical quality (peak-to-valley less than 150 nm and root-mean-square less than 25 nm for most devices). The process results in greater than 90% yield per wafer. The new mirrors increase concentric electrode count from three to four to provide more flexibility in controlling spherical aberration.

Experimental testing of these mirrors with a wavefront sensor verifies the aberration analysis. Finally, two in-depth imaging demonstrations (Blu-ray disc and confocal microscope) with these mirrors show the mirrors’ utility in imaging systems. This

17 dissertation is unique in that it consists of analytical aberration results, a novel electrostatic, deformable-membrane mirror, and experimental verification of the

 aberration analysis.

Dissertation Organization

Chapter 2 describes the novel design of a deformable mirror on a silicon-oninsulator (SOI) wafer. Chapter 3 contains the framework for the Hamiltonian optics analysis with presentation of the global and local coordinate systems. It also defines the characteristic aberration function. Chapter 4 analyzes the inherent aberrations and imaging performance of a single varifocal mirror. Chapter 5 experimentally verifies my

Chapter 4 analysis with the mirrors described in Chapter 2. Chapters 6 and 7 demonstrate these mirrors in two no-field imaging systems: an optical disk pick-up and a commercial confocal microscope.

18

CHAPTER 2

MEMS MIRROR DEVELOPMENT

Novel MEMS Mirror Designs

This chapter provides an overview of MEMs mirrors developed for my master’s degree as background and motivation for the mirrors developed in this dissertation. A wet-release and a dry-release process etched these single silicon wafer mirrors. A lowstress SU-8 2002 recipe was developed that reduces the required voltage for a given displacement of similar sized mirrors and air gaps. The second mirror design uses a silicon-on-insulator wafer to create a vertical etch stop for the dry-release process. It results in large deflections, high optical quality, and has high yield.

Wet- and Dry-release MEMS

Mirrors on Single Silicon Wafer

My master’s degree work developed simple electrostatic wet-release and dryrelease single wafer deformable mirrors. Optimizing the SU-8 processing parameters allowed for a total stroke of approximately 15

 m for a 3 mm by 4.24 mm elliptical-

boundary mirror at 320 V, limited by snapdown [1]. Figure 10 shows a picture (a), an

interferometric photograph (b), and a surface profile (c) of our dry-release mirrors [1].

The fixed-perimeter mirrors exhibited excellent optical quality with less than 50 nm peak-to-peak surface deviation when at rest. The simply supported mirrors (mirrors supported by thin 20% duty width strips at the perimeter) exhibited less than 100 nm

19 peak-peak. I briefly describe these mirrors as background to the refined mirrors developed as part of this dissertation.

140

100 100

(nm)

60

200

300

20

(a) (b)

200

(c)

400

Process 1

140

Figure 10: (a) Picture of two 2 mm x 2.82 mm elliptical-boundary mirrors mounted and

100

100

(a) wire-bonded on a TO5 header. (b) Optical interference micrograph of a 1 mm x 1.4 mm

(nm) elliptical boundary mirror. (c) Surface profile of a 1 mm x 1.4 mm elliptical-boundary mirror.

300

20

200 400

Process 1 Process 2

Two of the developed SU-8 2002 processes survived a wet-etch release in

(a) (b)

BOE 6:1 and tetramethylammonium hydroxide (TMAH). One process yielded membranes with half of the intrinsic stress as the other, but demonstrated poorer adhesion for long etch times in TMAH (limiting air-gap depth). The reduced stress allows twice the stroke for similarly sized devices at the same voltage potential. The intrinsic stress value met the lowest published SU-8 2002 intrinsic stress value for devices requiring etching of several microns of substrate, which was 13 MPa for a dry-release device [112].

To my knowledge, these mirrors are the only published large-deflection polymer mirrors released by wet etching, where wet etching allows processing of a large number of wafers at one time and has lower equipment costs than dry etching. Another one of the developed SU-8 processes benefited from use of a gentler dry-etch release in xenon difluoride and presents a simpler process that eliminates the need for two oxide layers

Process 2

(b)

Process 3

Process 3

20 and their associated masks. The dry etch resulted in greater gap depth without damage to the SU-8 film and prevents stiction of larger-dimension membranes.

For the dry-release process, xenon difluoride (XeF

2

) in a XACTIX etcher etched silicon beneath the membrane for release. A surface micromachining process in combination with an isotropic etchant (XeF

2

) created 10

 m peak-peak scalloping of the

bottom silicon counter electrode (Figure 11). This scalloping minimized their total stroke

capability and indicates local variation in etch depth. The silicon-on-insulator devices described in this dissertation mitigate these problems and create a controlled etch depth by means of an oxide etch stop.

Figure 11: A 200x scanning electron microscope image of a 1 mm fixed-perimeter, circular-boundary mirror after xenon diflouride etching with the top membrane removed.

The etch vias were above the centers of the numerous dark circles. The lighter gray areas are higher silicon peaks where the etch fronts merged [1].

Fixed-perimeter devices show 3 dB frequency responses (frequency at which magnitude decreases by 1/√2 from its maximum magnitude when actuated at low frequency) of 3-20 kHz with lower stress and larger diameter mirrors having the lowest frequency responses. Simply supported devices show 3 dB frequency responses of

500 Hz-20 kHz. The mirrors range from 1-3 mm in diameter and have circular boundaries for normal incidence to elliptical boundaries for

2

= 45 o

.

21

Dry-release MEMS

Mirrors on Single SOI Wafer

We describe here and in a previous paper [115] the use of a silicon-on-insulator

(SOI) wafer to utilize the buried oxide layer as a vertical etch stop. This ensures a flat

profile at a prescribed depth for the bottom of the air gap (Figure 12). Investigators have

previously used a single SOI wafer to fabricate scanning mirrors (tip/tilt) [116] and a process using more than one SOI wafer for focusing mirrors [117], but this is the first focusing mirror that we know of that requires only one SOI wafer. The simple process has greater than 90% device yield and results in high-optical-quality mirrors as shown below.

Figure 12 shows the design of the mirrors. First, we spin and pattern SU-8 2002

on an SOI wafer. Table 1 lists the steps performed for three different SU-8 2002

processes that we explored. We present in detail the steps of process-1. Initially, 10 min at 115 C on a Barnstead PMC DataPlate Digital Hot Plate 722A dehydrates the wafer.

SU-8 2002 is spun onto the wafer at 3250 RPM for 30 s. The wafer rests for 30 min at ambient to allow the solvent to evaporate [118]. Patterned exposure at 78 mJ/cm

2

defines

4

m vias in the SU-8 2002 for later release of the membrane (Figure 12). The post

exposure bake (PEB) consists of ramping from 45 o

C to 60 o

C at 1 o

C/min, holding at

60 o

C for 60 min, and allowing the wafer to cool on the hotplate. SU-8 developer develops the film in 15 s and isopropyl alcohol stops the process prior to a de-ionized water rinse. The wafer ramps from 45 o

C to 118 C at 3 o

C/min, holds at 118 o

C for

30 min, and cools on the hotplate for the hardbake step. Next, a metal lift-off process utilizing NR-1500PY Futurrex negative resist creates 5

 m vias in the top 6 nm chrome

22

(Cr) and 160 nm gold (Au) reflective surface and top electrodes. An additional 6 nm

Cr/200 nm Au is evaporated onto the backside of the wafer to provide good electrical contact with the silicon counter electrode. Dicing to 125

 m depth on the backside of the wafer prepares it for future separation of individual mirrors. Acetone and IPA remove

S1813 from the frontside. At 3 torr in a Xactix XeF

2

etcher the mirrors are released

(Figure 12 (right)) with 25 etch cycles for 30 s/cycle, 35 etch cycles for 20 s/cycle, and

360 etch cycles for 25 s/cycle. The initial 10-15 etch cycles remove any oxide that may have grown on the wafer. For the single silicon wafer mirrors, 30 s/cycle proved ideal.

For this reason, 30 s/cycle was tried first with these mirrors. SOI wafers have more exposed silicon on the sides of the wafers than a single silicon wafer. Furthermore, different masks for the two mirror designs resulted in a different amount of surface area of exposed silicon on the top of the wafers. One observes pressure in the chamber during etching to watch for it to stop increasing, where an increase in pressure indicates continuation of the exothermic reaction (2XeF

2

+ Si

2Xe + SiF

4

). Etching slows or stops when there is a lack of free fluoride in the chamber and it can no longer react with silicon. The reaction stopped early at 30 s/cycle, so 20 s/cycle was tried. The reaction did not slow or stop with 20 s/cycle, so 25 s/cycle was determined to be the best length of a cycle for the majority of the etching. Ideally, rest times should be used to prevent heating of the sample (as was done with the same process on silicon wafers), but were not possible due to time constraints. The temperature the mirrors reached during this process was not measured. If the samples reached high temperatures (due to the exothermic reaction), then it could have acted as an additional hardbake. It was noted that the

23 samples were cool enough to handle with standard nitrile gloves immediately after

release and purging of the chamber. Figure 13 shows the flatness (less than 90 nm peak-

peak deviation) of a process-1 released device with a tilted interferogram.

Table 1: Three SU-8 processes and their resulting intrinsic stress of the membrane.

Process #

Substrate

Process 1

SOI wafer with 1

 m oxide layer and 40

 m device layer

Process 2

SOI wafer with 2

 m oxide layer and 50

 m device layer

Process 3

SOI wafer with 2

 m oxide layer and 50

 m device layer

Membrane

Thickness

2.5

 m

Dehydration

115 o

C for 10 min

Softbake

Exposure

30 min room temperature

30 min room temperature

78.2 mJ/cm

2

45°C 1 min, 65°C

1 min, 80°C 3 min,

45°C

80 mJ/cm

2

PEB

Development

78.2 mJ/cm

2

45 o

C 0, ramp at

1 o

C/min to 60 o

C 1 hr, ramp at 1 o

C/min to 21 o

C

15 s

45 o

C 1 min, 65

85 o

C, 45

15 s o

C o

C 45 o

C 1 min, 65

85 o

C, 45

15 s o

C o

C

Hardbake*

45°C, ramp at 3 o

C/min to 118°C 30 min, 21 o

C ramp at 3 °C/min to o

135°C 30min, 21 C ramp at 4 °C/min to o

165°C 30min, 21 C

Etching in

Xactix XeF

2

25 etch cycles for

30 s/cycle, 35 etch cycles for 20 s/cycle, and

360 etch cycles for

25 s/cycle

20 etch cycles for

30 s/cycle and 410 etch cycles for

25 s/cycle

20 etch cycles for

30 s/cycle and 410 etch cycles for

25 s/cycle

Intrinsic

8.8 MPa 9.7 MPa 12.7 MPa

Stress

Note: Ramps between set temperatures are done at the automatic speed (approximately 6 of the hotplate, unless otherwise noted. For recipes ending at 45 o o

C/min)

C, the wafer is removed from the hotplate and cooled to room temperature on a clean aluminum surface. For example, for the process-2 PEB the wafer soaks for 1 min at 45 o

C, ramps at the automatic speed of the hotplate to

65 o

C, soaks at 65 o

C for 1 min, ramps at the automatic speed of the hotplate to 85 o

C, immediately begins cooling on the hotplate to 45 o

C, and is removed from the hotplate and cooled to room temperature on a clean aluminum surface.

*Temperature was not monitored during dry-release and an additional hardbake may have occurred.

24

Figure 12: (left) Cross section of the unreleased mirror. 2.5

 m of SU-8 2002 is photopatterned on an SOI wafer. 60

Å

chrome (Cr) and 160 nm gold (Au) are evaporated as a reflective coating and to serve as the top electrodes. 200 nm Au is evaporated onto the bottom side for the silicon counter-electrode. (right) 4

 m vias in the SU-8 and 5

 m vias in the gold layer allow for dry-etching of the silicon with xenon difluoride to create an air gap.

E4

E3

E2

E1

Figure 13: (left) Tilted interferogram showing the flatness of the process-1 3 mm x

3.011 mm mirror. (right) Top view with labels on the four electrodes.

Three different SU-8 processes yield different intrinsic stress in the film while still producing a quality film with well-defined (4

 m square) lithographic features.

Process 3 nearly repeats a process done for the single silicon wafer design and results in a similar intrinsic stress value of 12.7 MPa. I incorporate a softbake at room temperature

[118] for the other two processes to try to minimize stress and also for future design of the devices that may incorporate a lateral etch stop (discussed in more detail below), as it makes for a better film over surfaces with highly variable height structure. Process 2 also differs from process 3 by having a lower hardbake temperature of 135 o

C. Process 1 has a

25 lower hardbake temperature (118 o

C) than both processes 2 and 3. It also has a lower maximum PEB temperature of 60 o

C. Keller et al. [118] and Lukes et al. [1] suggest that lower baking temperatures at all stages lead to lower intrinsic stress levels. Process 2 and process 1 have intrinsic stress values of 9.7 MPa and 8.8 MPa, respectively. One can find the details of the intrinsic stress values below in the Device Characterization section.

The single silicon wafer design had two issues. One was the lack of a vertical etch stop, which these SOI mirrors mitigate with a buried oxide layer. The other issue was a lack of a lateral etch stop, which prevents precise control over the lateral dimension of the free membrane. Since xenon difluoride is an isotropic etchant and the etch vias lie in concentric elliptical patterns for elliptical-boundary mirrors, the lateral etch front expands at the same rate around the entire perimeter of the mirrors. This means that a 4.000 mm x

5.657 mm mirror with additional etching of 340

 m around the perimeter will have a

membrane boundary of 4.340 mm x 5.997 mm (Figure 13). The ratio of short axis to long

axis of the mirror surface is √2/2, whereas the ratio of the membrane is 0.724. Chapter 4 explains the importance of this ratio for ridding a system of parabasal astigmatism.

Ideally, the perimeter of the membrane should be 4.000 mm x 5.657 mm. In the future, the mirrors should incorporate a perimeter trench filled with a material such as oxide or silicon nitride to act as a lateral etch stop.

The lateral etch stop would benefit the fixed-perimeter mirrors we describe here.

It also would allow for simply-supported mirrors. Creating cutouts in the SU-8 2002 membrane around the perimeter reduces the percentage of the membrane that “fixes” the mirror to the perimeter, thus creating a simply-supported boundary condition. Without a

26 lateral etch stop, holes around the perimeter in the SU-8 cause great over-etching in the lateral direction. XeF

2

has better diffusion through the larger holes around the perimeter of the membrane compared with the small etch vias. According to Himmer [119], a 10% duty width support (SU-8 material) around the perimeter reduces the required pressure needed to deform the membrane by a factor of 2 and leads to more parabolic shape.

Furthermore, Shaw et. al [120] showed that the largest wavefront aberration in high-NA applications exists at the edge of the pupil, so having flexibility at the perimeter most likely leads to greater aberration compensation.

Device Characterization

This section includes static characterization of voltage versus center displacement, surface flatness, and Zernike fits of the mirrors’ surface shape under different voltage profiles. We also note that these MEMS mirrors are particularly designed for confocal applications or no-field imaging. They have periodic vias on the topside that diffract

635 nm and 433 nm light rays at 1.2

o

and 0.8

o

, respectively. A confocal system spatially filters these rays out and only a loss of irradiance of approximately 3% (5

 m vias spaced

30

 m apart or 5

2

/30

2 ≈3%) should be observed. I identify mirrors by process-#, short-axis diameter, and incidence angle. For instance, a process-1 4 mm 45 o

mirror has a short axis of 4.000 mm and a long axis of 4.000/cos(45 o

) = 5.657 mm for imaging at

2

= 45 o

.

27

Displacement due to Electrostatic Pressure

The sag of the mirror surface, assuming

C a2

=

in Equation 13

may be expressed as,

, (1) where

ɛ

1

,

ɛ

2

, and

ɛ

3

, are the relative permittivities of SU-8, air, and thermal oxide, respectively. The details of this equation may be found in Appendix D. Variables

a

,

b

,

h

1

,

s

o

, and

h

3

are the minor semi-axis of the ellipse, major semi-axis of the ellipse, thicknesses of SU-8, the air gap depth, and thickness of thermal oxide, respectively.

V

is the voltage. Tension,

T,

is related to the intrinsic stress,



of the membrane as follows,

T =



h

1

. The intrinsic stress values in Table 1 are calculated with values of 3.9 [121] and

4.1 [114] for the relative permittivity of thermal oxide and SU-8 2002, respectively. The average of linear fits of the voltage applied squared, divided by small displacements

(< 5

 m of the metal region) of several mirrors determined these values.

Phase shift interferometry (

= 850 nm) measures the deflection of these mirrors

(Figure 14). Because no lateral etch stop halts the isotropic etchant, over-etching occurs.

The voltage versus center deflection data shows displacement of only the optical surface and not total displacement of the membrane including around the perimeter of the mirror

surface (Figure 15). For example, the process-1 4 mm 45

o

=

2

mirror (shown in Figure

15) has an optical surface that extends 4.000 mm x 5.657 mm, whereas the free-standing

membrane extends 4.340 mm x 5.997 mm. Overall, smaller mirrors, larger air gaps, and

28 higher intrinsic stress mirrors require more voltage for a given displacement. Snapdown into a 50

 m air gap limits the maximum mirror sag to 22

 m.

Figure 14: A process-3 4 mm X 5.66 mm mirror with 466 V and a maximum deflection of 22

 m (

= 850 nm).

25

20

15

10

5

Process-1 3 mm 5 o

=

2

Process-1 4 mm 45 o

=

2

Process-2 3 mm 5 o

=

2

Process-2 4 mm 45 o

=

2

Process-2 5 mm 0 o

=

2

Process-3 4 mm 0 o

=

2

Process-3 4 mm 5 o

=

2

Process-3 4 mm 10 o

=

2

Process-3 4 mm 45 o

=

2

0

0 50 100 150 200 250 300 350 400 450 500

Voltage (V)

Figure 15: Voltage versus deflection for several mirrors. The perimeters of the mirrors experienced lateral over-etching of 100-200

 m, which makes interpretation of their voltage profiles complex. One can observe that smaller mirrors, larger air gaps, and higher intrinsic stress mirrors require more voltage for a given displacement. Snapdown into a 50

 m air gap limits the maximum displacement to 22

 m.

29

Dynamic Characterization

Larger and lower stress devices have the lowest resonant frequencies. For this reason, magnitude and phase plots are shown for one of the largest and lowest stress mirrors released: a fixed-perimeter, process-2, 5 mm 0 o

mirror (Figure 16). A dc voltage

with a small, superimposed sinusoidal driving voltage actuates the mirror. The mirror exhibits a 3-dB frequency of 830 Hz and a 90 o

phase lag greater than 5 kHz, as observed by reflecting light off the mirror at a small angle and measuring light intensity through a pinhole near focus.

-8

-10

-4

-6

0

-2

X: 824.1

Y: -2.915

0

-20

-40

-60

-12

10

0

10

1

10

2

Frequency (Hz)

10

3

10

4

-80

10

1

10

2

10

3

Frequency (Hz)

10

4

Figure 16: (left) Normalized magnitude plot of a process-2 5 mm 0 o

mirror with 3-dB response shown. (right) Phase response of a process-2 5 mm 0 o

mirror.

Surface Shape

Table 2 and Table 3 show Zernike defocus coefficient,

𝑍

; Zernike 3 rd

order spherical coefficient,

𝑍

; and Zernike 5 th

order spherical coefficient,

𝑍

, for different voltage profiles (limited to a maximum of 120 V difference between electrodes to prevent

electrical breakdown) on two different mirrors. Figure 13 defines electrodes E1-E4. We

determine Zernike coefficients by imaging the MEMS mirror at normal incidence with a

30 phase-shift interferometer. The elliptical boundary is scaled to a circular boundary and

ZERNFUN2.m [122] subsequently fits the Zernike polynomial out to 36 terms (see

Appendix C). In Table 2, a process-1 3 mm 5

o

mirror shows a maximum positive primary spherical aberration of 157 nm at 6.5

 m deflection and a minimum negative of -652 nm at 10

 m deflection. This is a total positive to negative range of 809 nm of primary spherical aberration. The mirror has a range of 324 nm secondary spherical aberration for displacements between 5.9

 m and 8.1

 m. A process-2 5 mm 0 o

mirror also shows a range of 1

 m of primary and 450 nm secondary spherical aberration over 5.0-7.0

 m deflection (Table 3). (Note: The 5.000 mm mirror requires lesser voltage for the same displacement.) The spherical aberration correction observed is for these particular families of voltages. It is illustrative of the range of control available at these deflections, which are between 20% and 50% of the maximum deflection. They do not represent an exhaustive characterization of the range of shapes available to the mirrors.

These mirrors typically show excellent flatness. A process-1 4 mm 45 o

mirror

exhibits 147 nm peak-to-valley and 22 nm RMS when flat (Figure 17). Often, smaller

diameter mirrors (such as 2-3 mm circles) show the best flatness. This large mirror demonstrates excellent flatness and is representative of the majority of the mirrors

released. Figure 18 shows similar flatness for a process-1 4 mm 5

o

mirror.

31

Table 2: Change in Zernike defocus and spherical aberration coefficients for a process-1

3 mm 5 o

mirror.

E1 E2 E3 E4

(nm) (nm) (nm)

(V) (V) (V) (V)

(nm)

450

450

450

330

450

450

330

330

330

330

450

330

330

330

330

450

330

450

450

330

330

330

450

330

330

450

450

330

330

330

450

450

330

450

330

330

330

450

330

330

450

450

6153

7008

7578

8637

7717

8092

4261

4980

5877

5927

5313

6458

7664

6794

450

330

330

450

450

450

330

450

330

330

330

450

330

450

2786

3335

3552

4192

3576

3727

2054

2340

2876

2956

2487

3084

3788

3275

-144

-103

-485

-234

-84

-468

-42

71

91

-247

-207

157

-207

-181

193

7

122

-38

91

214

30

82

-110

-5

101

-48

-68

73

450 450 450 330 9929 4588 -519 73

450 450 450 450 10297 4790 -652 110

Table 3: Change in Zernike defocus and spherical aberration coefficients for a process-2

5 mm 0 o

mirror. The mirror had some charge build-up, which required greater voltages for the same displacement (less than 20 V higher than if the mirror had no charge buildup during testing).

E1

(V)

320

E2

(V)

320

E3

(V)

320

E4

(V)

320

(nm)

9978

(nm)

4993

(nm)

-232

(nm)

31

200 320 320 320 7699 4094 108 -131

320

320

320

320

200

200

320

200

320

200

200

320

320

320

320

200

200

320

200

200

320

200

320

200

320

320

200

200

320

200

320

320

200

200

200

320

320

320

200

200

6832

7561

8871

6493

7004

5358

4942

5779

6070

2927

3319

3579

4418

3208

3728

2705

2232

2897

3015

1429

182

-368

-395

-567

-67

404

-67

-89

-31

8

53

218

-9

134

-204

-98

236

42

-29

5

32

P-V = 147 nm; RMS = 22 nm

80

20

40

60

80

100

120

140

60

40

20

0

-20

160

-40

180

-60

200

50 100 150 200

Figure 17: (left) Surface profile of a process-1 4 mm 45 o

mirror scaled to a circular perimeter as measured by phaseshift interferometry. (right) a photograph of the mirror tilted in the interferometer for a flat (no voltage applied) process-1 4 mm 45 o

(4.000 mm x 5.657 mm) mirror. Lateral under-cutting extends the membrane beyond the optical surface to 4.340 mm x 5.997 mm. This relatively large MEMS mirror exhibits excellent optical quality with 147 nm peak-to-valley and 22 nm RMS.

P-V = 113 nm; RMS = 11 nm

20

50

40

40

30

20

60

10

80

0

100

-10

120

-20

140

-30

160

-40

180

-50

200

50 100 150 200

Figure 18: (left) Surface profile of a process-1 4 mm 5 o

mirror scaled to a circular perimeter as measured by phaseshift interferometry. (right) A photograph of the mirror tilted in the interferometer for a flat (no voltage applied) process-1 4 mm 5 o

mirror. The

MEMS mirror exhibits excellent optical quality with 113 nm peak-to-valley and 11 nm

RMS.

33

Figure 19 shows the same flat mirror as in Figure 17, but with the surface shape

reconstructed from Zernike terms. It also provides the Zernike spectrum with all terms above 5 magnified by 10. Most Zernike terms are below 8 nm. A process-1 4 mm 5 o

mirror also has excellent flatness, as shown in Figure 20. The diagram has Zernike

spectrums for this mirror with 5.0

 m and 10.9

 m displacement, which correspond with

data presented in Chapter 5. Figure 21, Figure 22, and Figure 23 compare Zernike terms

for a mirror with equipotential voltage and the voltages varied to minimize spherical aberration. The purpose of showing these surface shapes is purely to aid in understanding the results in Chapter 5. Equipotential voltages demonstrate the mirrors’ imaging capabilities if they only had one electrode or one high voltage supply. One can note that they maintain fairly parabolic shape up to 10

 m deflection. At 10

 m deflection they show primary spherical aberration of 200-300 nm and secondary of 20 nm with equipotential voltages. At small deflections of 5

 m, they show even less primary spherical aberration of 50-70 nm and secondary of 10 nm with equipotential voltages. By varying the voltages, the secondary and primary spherical aberrations are reduced to nearly 0 nm.

34

10x

100

50

0

-50

-2

-1

0

1

2

-2

-100 Defocus

-150

-200

0

1

5

2

10

Primary spherical

15 20

(a)

0.2

0.15

0.1

0.05

0

(

 m)

-0.05

-0.1

-0.15

-0.2

-2

-1

0

1

25

2

-2 -1 0 x (mm)

(b)

Secondary spherical

30 35

-1 0 x (mm)

(c)

40

1 2

-0.5

-1

-1.5

1

0.5

0

(

 m)

Figure 19: A flat (no voltage applied) process-1 4 mm 45 o

mirror. (a) Zernike terms as generated from a phase-shift interferometer. Terms above Z5 are magnified by 10x.

Defocus, primary spherical, and secondary spherical are Z5, Z13, and Z25, respectively.

(b) Zernike terms shown in (a) reconstruct the wavefront. (c) The same wavefront as in

(b), but on a scale used in Chapter 5 for measured wavefront sensor results to allow for comparison. This data is for physical deflection of the mirror when viewed at normal incidence. For direct comparison with Chapter 5 data, this surface data needs to be multiplied by 2*cos(

 incidence.

2

) to convert the surface height to appropriate wavefront at 45 o

35

0 V

-2

-1

0

1

2

-2 -1 0 x (mm)

1 2

0.5

0 (

 m)

-0.5

150

100

50

0

-50

-100

-150

-200

0 5 10 15 20 25

Zernike Mode Number

30 35 40

200 V

-2

-1

0

1

0.5

0 (

 m)

1000

500

0

-500

-1000

-1500

-2000

-2500

-3000

0

2

-2 -1 0 x (mm)

1 2

-0.5

10 20

Zernike Mode Number

30 40

-2 0.5

4000

2000

-1

0

0

0

(

 m)

247 V

-2000

1

-4000

2

-2 -1 0 x (mm)

1 2

-0.5

-6000

0 10 20

Zernike Mode Number

30 40

Figure 20: Wavefronts and Zernike coefficients for a process-1 4 mm 5 o

mirror used in

Chapter 5. All Zernike modes above 6 are 10x. The surface heights are reconstructions without defocus, tip, tilt, and piston (modes 1, 2, 3, and 5), but include all other terms.

With an equipotential voltage of 0 V, the mirror is flat. An equipotential voltage of 200 V yields 5

 m mirror sag. The mirror sag is 10.9

 m at 247 V.

36

3000

2000

1000

0

-1000

-2000

-3000

-4000

-5000

0

=10

 m equipotential

=10

 m spherical minimized

Defocus

Primary spherical

Secondary spherical

5 10 15 20

(a)

25 30 35 40

-2 -2

1 1

-1

0

0.5

0

(

 m)

-0.5

-1

-1

0

0.5

0

(

 m)

-0.5

-1

1 1

-1.5

-1.5

2

-2 -1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

(b) (c)

Figure 21: Comparison of Zernike terms and wavefronts for a process-1 4 mm 45 o

mirror with equipotential (233 V) and varied voltages to minimize spherical aberration terms

(E1 = 234.5 V, E2 = 198 V, E3 = 255 V, and E4 = 285 V). For both voltage profiles the center of the mirror is displaced 10

 m. (a) Zernike terms as generated from a phase-shift interferometer with normal incident light rays on the mirror. Terms above Z5 are magnified by 10x. Primary and secondary spherical aberration decrease for the varied voltage case, where the varied voltages reduce the primary spherical aberration by approximately 300 nm. (b) Equipotential wavefront. (c) The wavefront with spherical aberration minimized.

37

1000

500

0

-500

-1000

-1500

-2000 equipotential spherical minimized

Defocus

Primary spherical

Secondary spherical

-2500

0 10 20

(a)

30 40

-2 -2

-1

0

1

0.5

0

(

 m)

-0.5

-1

-1

0

1

0.5

0

(

 m)

-0.5

-1

1 1

-1.5

-1.5

2

-2 -1 0 x (mm)

(b)

1 2

2

-2 -1 0 x (mm)

(c)

1 2

Figure 22: Comparison of Zernike terms and wavefronts for a process-1 4 mm 45 o

mirror with equipotential (184.5 V) and varied voltages to minimize spherical aberration terms

(E1 = 195 V, E2 = 173 V, E3 = 194 V, and E4 = 206 V). For both voltage profiles the center of the mirror is displaced 5

 m. (a) Zernike terms as generated from a phase-shift interferometer with normal incident light rays on the mirror. Terms above Z5 are magnified by 10x. Primary and secondary spherical aberration decrease for the varied voltage case, where the varied voltages reduce the primary spherical aberration by approximately 700 nm. (b) Equipotential wavefront. (c) The wavefront with spherical aberration minimized.

38

0

1

-2

-1

10x

1000

500

0

-500

-1000

-1500

-2000

-2500

-3000

0

Primary spherical

=5

 m equipotential

=5

 m spherical minimized

Secondary spherical

5

Defocus

10 15 20

(a)

25 30 35 40

0.5

-2

-1

0

(

 m) 0

1

0.5

0

(

 m)

2

-2 2

-0.5

2

-2

-0.5

-1 0 x (mm)

(b)

1 -1 0 x (mm)

(c)

1 2

Figure 23: Comparison of Zernike terms and wavefronts for a process-1 4 mm 5 o

mirror with equipotential (200 V) and varied voltages to minimize spherical aberration terms

(E1 = 200 V, E2 = 185 V, E3 = 205 V, and E4 = 225 V). For both voltage profiles, the center of the mirror is displaced 5

 m. (a) Zernike terms as generated from a phase-shift interferometer with normal incident light rays on the mirror. Terms above Z5 are magnified by 10x. Primary and secondary spherical aberration decrease for the varied voltage case, where the varied voltages reduce the primary spherical aberration by approximately 500 nm. (b) Equipotential wavefront. (c) The wavefront with spherical aberration minimized.

39

CHAPTER 3

FRAMEWORK FOR HAMILTONIAN OPTICS ANALYSIS

Coordinate System and Aberration Coefficients

My analysis of aberrations is based on Buchdahl’s approach in

An Introduction to

Hamiltonian Optics

[108] and Dickensheets’ “Imaging performance of off-axis planar

diffractive lenses” [109]. Figure 24 (a) shows the global coordinate system with the

origin

O

at the center of the mirror plane and point

Q

lying on the mirror surface. Both object and image are in a homogenous medium with refractive index

n

. The object plane lies normal to the object-space, base-ray and contains point

P

10

. The image plane lies normal to the image-space, base-ray and contains point

P

20

. The object and image planes have local (

u

1

, v

1

) and (

u

2

, v

2

) coordinates, respectively (Figure 24b). The

following transformation relates

u

1

, v

1

, and

r

1

to the global mirror plane coordinate system,

. (2)

The following transformation relates

u

2

, v

2

, and

r

2

to the global mirror plane coordinate system,

. (3)

The aberration function is specified in terms of the exit pupil coordinates (

x

,

y

) and the image coordinates (

u

2

,

v

2

), where the exit pupil is chosen in a plane perpendicular

40 to and includes the origin. Mapping the optical path difference (OPD) variation onto a plane perpendicular to the base ray allows direct comparison of aberration terms to other standard optical elements and is convenient for Fourier analysis to predict the diffraction properties of the beam in the image plane. The pupil plane, therefore, lies parallel to the image plane, has its

y

-axis coincident with

Y

, and has a center at point

O

in

the mirror plane. Figure 25 shows the mirror plane (

X

,

Y

), pupil plane (

x

,

y

), and image plane (

u

2

,

v

2

). A dotted line illustrates the projection of a ray from point

P

2

in the image plane through point

Q

on the mirror surface and onto point (

x

,

y

) in the pupil plane. The mirror surface varies along

Z

as a function of

X

and

Y

according to

Z mirror

=

f(X, Y)

(Figure 26).

X

X

Mirror

Plane

Y

Q

P

10

O

P

20

Mirror

Plane

Z

Y

O

1 r

1

P

1 v

1 u

1

P

10

Z

(a)

(b)

Figure 24: (a) Global coordinate system with origin

O

at the center of the mirror plane.

The object plane contains point

P

10

and lies perpendicular to the object-space, base-ray

. The image-space, base-ray image plane also contains point

lies perpendicular to the image plane where the

P

20

. (b) Local object (

u

1

, v

1

) coordinate plane perpendicular to .

41

Restricting our analysis to mirrors, variables

r

1

and

r

2

must be real. The law of reflection requires that

2

= -

1

, where <

1 <

(Figure 26a).

X x

Pupil

Plane

Mirror

Plane

Q

(x, y)

O

P

2 u

2

Image

Plane

v

2

P

20

2

Y, y

Figure 25: Local exit pupil (

x, y

) coordinate plane for determination of aberration coefficients. The exit pupil plane contains the origin

O

of the mirror plane and is normal to the image-space, base-ray .

Mirror Plane

X

Mirror

Surface

Base Ray

r

1

1 r

2

2

= -

1

Z mirror

= f(X,Y)

Z

Mirror Plane

X

Pupil

Plane

Chief Ray

Q

O x u

1

Object Plane

P

1

P

10

q q

-

1

1 u

2

Base Ray

P

20

P

2

Z

Image Plane

Z mirror

= f(X,Y)

(a) (b)

Figure 26: (a) The law of reflection restricts the entrance and exit angles of the base rays

r

1

and

r

2

with respect to the varifocal mirror to be equal in magnitude and opposite in sign. (b) Detailed drawing depicting the chief ray and ray of interest through point

Q

for the -function.

42

Neglecting the finiteness of the wavelength and polarization, we develop an aberration function of the optical system in terms of the image plane and exit pupil variables. The magnification terms describe the linear relationship between the transverse object and transverse image coordinates in the Gaussian imaging approximation according to

(4)

The aberration function calculates the difference between the optical path length for a ray from point

P1

to point

P2

through point

Q

on the mirror relative to the optical path length of the chief ray through the origin (point

P1

to point

P2

through point

O

on the mirror),

. (5)

A positive -function indicates a longer optical path length for the specified ray than the chief ray. The sign(

r

1

) and sign(

r

2

) allow for virtual objects (-

r

1

) and images (-

r

2

) to be

analyzed. Figure 26b provides a more detailed view of the mirror, object plane, and

image plane. It shows the reference chief ray as a solid purple line and the comparative ray through point

Q

in the mirror as a dotted purple line.

We introduce normalized pupil coordinates and mirror displacement variables,

(6) where

a

is the minor semi-axis of an elliptical mirror (Figure 30). The major semi-axis is

b

. The center displacement of the mirror,



is equal to

f

(0

,

0). One might note that

x

and

y

43 are pupil plane coordinates, where the pupil plane is parallel to the image plane. For the case of , the rays have a circular projection with radius equal to the minor semiaxis of the elliptical mirror,

a,

through the pupil plane. This holds regardless of angle of incidence,

2

(as is further discussed in Chapter 4). Most of the analysis will be restricted to this case and, consequently,

x

and

y

are normalized to

a

for easier comparison with other published results that follow a standard convention of a normalized exit pupil.

Expanding the aberration function into a Taylor series about 0, the terms of order one in the pupil- and image-plane coordinates of the aberration function may be expressed as (pp. 26-30 in [108])

(7) where and represents terms of order

n

and higher. Coincidence with the base ray occurs for the limit in which

u

2

,

v

2

, , and go to zero. Because the exit pupil plane and the image plane lie perpendicular to the base ray, the aberration function has no linear dependence on pupil- and image-plane coordinates and are zero.

Aberrations may be discussed in terms of rays or wavefronts. Ray aberrations refer to the height of a ray where it intersects the image plane. The height of a ray is related to the local slope of the wavefront, which is the local first derivative of the wavefront. This indicates that if the wavefront is of order 2 in the pupil plane coordinate, then the ray height is one order less, or 1. The orders of aberrations in this dissertation are in terms of rays. The parabasal coefficients, terms of order two in the pupil- and image-plane coordinates, are defined as follows,

44

(8)

Due to plane symmetry about

Y

= 0, the coefficients of terms for which the sum of powers in and

v

2

are odd must be zero and the expression reduces to

(9)

The coefficients and prescribe defocus for the tangential and sagittal rays, respectively. If , then the two astigmatic planes meet. If , then no defocus exists at the image plane. We express the second-order aberration terms or terms of order three in the pupil- and image-plane coordinates as

45

(10)

The coefficients have a cubic dependence on pupil coordinates, so they will behave similarly to coma. Unlike 3 rd

order coma, however, they have no dependence on image height and the aberration is constant over a finite field-of-view. The terms

are quadratic in the pupil coordinates and linear in the field

are distortion variables thus providing linear astigmatism, and terms (wavefront tilt). Due to symmetry, we reduce this to

(11)

The third-order aberration terms are defined as follows,

46

(12) where aberration,

are independent of the field variables thus representing spherical

are cubic in pupil coordinates thus providing coma, and

are quadratic in the pupil coordinates thus contributing astigmatism, and are distortion terms.

Stop Shifting

Figure 27 shows the chief ray

CR p

that arises from a shift in the stop

p

from the mirror plane, where

CR

refers to the chief ray through the center of the mirror plane.

Using the geometrical relationships and , where and , the corresponding aberration function for a stop shift may be determined.

Using relationships described by Mahajan on pages 12-13 [110], the new aberration function may be expressed as,

(13)

47

Pupil Plane

Centered at

Mirror Plane

(x, y)

(x

1

, y

1

)

Image Plane

(u

2

, v

2

)

CR p

CR p

(+)

r

2

(+)

Figure 27: A change in position of stop by distance

p

changes the reference chief ray from

CR

to

CR p

for calculation of the aberration function.

After transforming to the new coordinate system, substitution of and

provides normalized coefficients in the new exit pupil plane. Equations (9),

(11), and (12) still represent the aberration terms, where are replaced with . For clarity, we subscript the coefficients determined with a stop shift from the mirror plane with a

p

.

Implementation Details

We analytically calculate the -function in Matlab by assuming a specific MEMS mirror shape in

Z

-direction. After performing a Taylor series expansion, specific terms of a given order are found by differentiation. For instance, the x

4

terms may be found by differentiating with respect to

x

four times and then setting

x

,

y

,

u

2

, and

v

2

equal to zero to rid the results of higher order terms. Appendix A explains in more detail how aberration terms are extracted from the entire -function. Appendix A also provides verification of my code by comparing the results of the -function with optical path difference plots in

Zemax. Appendix B compares our expansion including only 2 nd

and 3 rd

order terms with

48 the entire -function for a parabolic MEMS mirror to verify accuracy. It shows less than

2% difference between the two, thus indicating that higher-order terms are not warranted.

The distances

r

1

,

r

2

, and the radius of the exit pupil are in millimeters. The deflection of the mirrors should be less than 30

 m. A slight approximation exists in that the position of the ray across the mirror in

X

and

Y

coordinates assumes the mirror is flat

(i.e., it does not account for the slight shift in

X

and

Y

when

Z mirror

≠0). This introduces slight displacement in the pupil plane coordinates from the actual ray crossing at (

x’, y’

) to (

x, y

) for the aberration function (Figure 28). For an idea of the magnitude of error

introduced because of this approximation, if a ray passed through the image and pupil planes at 60 o

, then the maximum error in pupil coordinates

x

and

y

for a 3 mm diameter mirror at 30

 m deflection (maximum of

Z mirror

) would be 1.7%.

Q

X x

(

x’, y’

)

P

1

(

x, y

)

Z

P

2

Y

Figure 28: The actual ray from point

Q

to

P

2

crosses the exit pupil at (

x

’,

y

’), but our analysis provides the ray location as (

x

,

y

).

49

Definition of Surface Shape for Elliptical-Boundary Mirrors

Our research group is developing deformable membranes as variable-focus lenses.

We have suggested the use of elliptical-boundary membrane mirrors for light rays at angles other than normal incidence. Fabrication of the mirror surface on a wafer leads to a planar boundary such that the perimeter of mirror surface

Z

(

X, Y

) must be at constant displacement value Z =

(Figure 29).

Z

Y

Mirror Surface

Z

(

X, Y

)

Surface of

Silicon Wafer

X

Elliptical

Perimeter

Figure 29: A typical elliptical-boundary mirror, where the perimeter of surface

Z

(

X, Y

) must be at a constant displacement value

.

We represent

Z

as a polynomial in

X, Y

, with coefficients constrained so that

Z boundary

=

. The relationships and

map an ellipse to a circle (Figure 30).

Only even powers in

𝑋̃

,

𝑌̃

are included due to an assumption of symmetry. We define

2 nd

- and 4 th

-order even polynomial (

p

) equations as follows,

50

and (14)

~

Y a

Y b

X

~

X

Figure 30: Mapping an ellipse with semi-minor axis

a

and semi-major axis

b

in coordinates onto a normalized circle in coordinates.

(15)

51

CHAPTER 4

IMAGING PERFORMANCE OF A SINGLE MEMS MIRROR

An elliptical-boundary membrane mirror under uniform pressure across its surface naturally deflects with a paraboloidal shape in the

Z

-direction. Substituting

C a2

=

into Equation (14) from the previous chapter provides the simplest representation of this shape, , which will be used for the entire analysis in this chapter. The variables

r

1

,

r

2

,

,

a

, and

b

are real, where a≠0.

Parabasal Coefficients

Beginning with the parabasal coefficients, four coefficients are zero due to plane symmetry, as explained in the previous chapter. All other coefficients equate to zero, except

(16)

For no astigmatism,

b

1

and

b

5

must be equal. If the imaging equation is satisfied, both

b

1 and

b

5

will be zero. This requires that

b

=

a

/cos(

2

) (Figure 31) and

, the imaging equation, be maintained. We define the focal length of the mirror as .

These two relationships are upheld for the entire analysis unless otherwise noted. The imaging equation provides the following relationship for the distance to the image plane,

52

. (17)

MEMS

Mirror

(a)

1

Incident

Ray

MEMS

Mirror

(b)

1

2b

2a

1 b = a / cos(

1

)

(c)

Figure 31: For the two astigmatic focal lines to coincide in

Z

,

b

=

a

/cos(-

2

) =

a

/cos(

1

) must be maintained. Alternatively, the diameter of the major-axis, 2

b,

must project as the same diameter of the minor-axis, 2

a,

at a given angle of incidence. (a) MEMS mirror shown with a flat surface and with center deflection

. The incident light ray approaches the mirror at an angle

1 to the normal of the mirror. (b) An inserted triangle with the hypotenuse representing the diameter 2

b

of the MEMS mirror surface. (c) The relationship between the length of the surface normal to the incident ray 2

a

and the diameter of the mirror surface 2

b

.

The aberration coefficients use the definition of

F

-number,

. Table 4

provides values of

r

2

and

F

for several deflections and incidence angles for the case of

. We present these values, because one should consider their coupled nature when interpreting the following aberration coefficients and plots.

53

Table 4: Corresponding natural focus at distance, deflection,



and incidence angle,

2

, for and

r a

2

, and

=

F

-number,

2 mm.

F,

for a given

(

m)

2

=

0 o

r

2

(mm)

2

=

20 o

2

=

45 o

2

=

0 o

F

(mm)

2

=

20 o

2

=

45 o

5

10

15

20

25

30

200

100

66.7

50

40

33.3

213

106

70.9

53.2

283

141

94.3

70.7

50

25

16.7

12.5

43.3 56.6 10

35.5 47.1 8.3

Second-Order Aberrations

53.2

26.6

17.7

13.3

10.8

8.9

70.7

35.4

23.6

17.7

14.1

11.8

Deferring to Buchdahl’s and Hamilton’s preference for describing the order of aberrations in terms of ray aberrations, I define the orders of aberrations in the same manner. The 2 nd

-order aberrations are defined as

Applying the two constraints for , to rid the system of parabasal astigmatism and defocus, the 2 nd

-order aberration terms with

variable stop location

,

p

, are

54

(18)

One should note that is defined with reference to the mirror only and not the optical system, where

F

for the optical system would be divided by the stop diameter. All terms go to zero for a normal incidence circular-boundary mirror (

2

= 0). The fieldindependent terms, 𝑐̃

1

and 𝑐̃

5,

have no dependence on stop location.

The linear astigmatism term, 𝑐̃

7

, goes to zero for stop at position

p

= 2

f

, where the physical aperture would most likely lie in object space. At

p

= 2

f

, the other two linear astigmatism terms are equal, 𝑐̃

3

= 𝑐̃

12

. This only occurs for a particular deflection, or

f,

of the mirror. The resulting image plane is tilted in the

Z

direction with linear dependence on

u

2

.

For an object at infinity (

55

), Equation (17) reduces to . Using the relationships , it may be shown that . The normalized

c

terms for an

object at infinity with a variable stop location

may be expressed as,

(19)

All of these terms go to zero for normal-incidence angle (tan(0) = 0).

Placing the stop at mirror surface, or

p

= 0, omits three distortion terms from the general case of 2 nd

-order terms with variable stop location (Equation (18)).

For stop at mirror

, the coefficients reduce to

56

(20)

The 𝑐̃

1

and 𝑐̃

5 coefficients remain constant across the field-of-view in the image plane; therefore, I refer to them as 2 nd

-order, field-independent terms. The 𝑐̃

12

and 𝑐̃

3

coefficients result in an astigmatic aberration. The 𝑦̃

2

𝑥̃

2

astigmatic shape in the exit pupil is linearly proportional to the field location

u

2

. The 𝑐̃

7

coefficient also represents an astigmatic aberration, where the 𝑦̃𝑥̃

shape in the exit pupil is linearly proportional to the field location

v

2

. All of the 2 nd

-order ray aberrations go to zero for a normal angle of incidence mirror.

For an object at infinity ( ) and stop at mirror

, the terms simplify to

(21)

One might note that off-axis paraboloidal (OAP) mirrors introduce no spherical

aberration when used for infinite-conjugate imaging (Figure 32). The use of an elliptical-

57 boundary deformable mirror with parabolic shape at a given angle of incidence differs from using an OAP mirror, however, because it has two 2 nd

-order spherical terms, 𝑐̃

1

and 𝑐̃

5, that are non-zero. An off-axis paraboloidal mirror is not symmetric about

X

= 0. For simplicity of actuation and control, our research group is unique in that we use symmetric electrodes, which allow for control of spherical aberrations. This means the ellipticalboundary mirrors described in this dissertation are symmetric about

X

= 0. The 2 nd

-order, field-independent terms, 𝑐̃

1

and 𝑐̃

5, represent a type of spherical aberration, since they only depend on exit pupil coordinates. They are different from the most commonly discussed 3 rd

-order spherical aberration in that they have cubic dependence on pupil

coordinates. They manifest more similarly to coma for this reason (Figure 32). They are

the dominate shape difference between a deformable off-axis mirror that is symmetric about

X

= 0 and the ideal infinite-conjugate OAP mirror shape for perfect imaging.

Dynamic paraboloidal

MEMS mirror

-2 0.1

-2

Off-axis paraboloidal mirror from parent on-axis paraboloidal mirror

0.1

-1

0

0.05

0 (

 m)

-0.05

-1

0

0.05

0

(

 m)

-0.05

1 1

2

-2 -1 0 x (mm)

1 2

-0.1

2

-2 -1 0 1 2

-0.1

x (mm)

Figure 32: For infinite-conjugate imaging with zero field, an off-axis paraboloidal (OAP) mirror introduces no aberration to the system. A dynamic MEMS mirror with plane symmetry introduces a small amount of 2 nd

-order, field-independent aberration that presents similar to coma. The wavefronts shown have

r

2

2

= 45 o

mirror.

= 150 mm or

= 9.4

 m for an

58

Although the case

r

1

=

r

2

may not be completely practical in the sense that these mirrors have dynamic focus control, having an understanding of what the best midpoint condition might be for mid-range performance of a given device may improve ease of implementation of these devices in optical systems. For instance, root-mean-square

(RMS) graphs later in this chapter will show a plateau in aberrations for no-field conditions as the mirror deflects through

r

1

=

r

2

. It limits variation in RMS values across a range of deflections of the device, which should create a more uniform image than a more varying range of RMS values over a given deflection range. Furthermore, these types of mirrors offer high-speed focus control where a small focal range may be sufficient for a given application. This plateau in aberrations near

r

1

=

r

2 appears to be greatest for no-field conditions. For the object and image at equal distances from the mirror plane ( ), Equation (17) reduces to and , which are both twice the values as for the infinite-conjugate case. We may express the normalized

c

-terms

for stop at mirror and r

1

=

r

2

(object and image at equal distances) as,

(22)

For this case, no field-independent, 2 nd

-order aberration exists due to symmetry about plane

X

= 0.

59

Peak-to-valley, 2 nd

-Order

Aberration Terms for an Object at Infinity

The peak-to-valley, 2 nd

-order aberration terms for different angles of incidence

and displacements of the mirror are shown in Figure 33. The plots shown in (a) present

astigmatism with a linear dependence on field variables (values shown are for atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

). The left plot shows peak-to-valley astigmatism aberration for a field of atan(

u

2

/

r

2

) = 1 o

with a mirror at

2

= 2.5

o

and

= 20

 m (the displacement achievable by the SOI mirrors presented in this dissertation) to be 61 nm. A peak-tovalley aberration of 61 nm maintains

/10 imaging for 633 nm wavelength light.

Increasing the incidence angle to only 5 o

doubles the peak-to-valley aberration to 122 nm.

This indicates that using these mirrors at large deflections results in large astigmatism values as the incidence angle increases even at small field angles (1 o

). However, if one desired to only use 5

 m of deflection, then the mirror could be used at any angle of incidence while still maintaining

/10 imaging for field angles of 1 o

.

Field-independent aberration, shown in Figure 33b, has a peak-to-valley of 70 nm

for a mirror at

2

= 2.5

o

and

= 20

 m. It does not maintain

/10 imaging for 633 nm wavelength light. Increasing the incidence angle to 5 o

, approximately doubles the peakto-valley aberration to 134 nm. These values slightly exceed the values for astigmatism.

As it does for 2 nd

-order astigmatism, the mirror maintains

/10 imaging at any angle of incidence with less than 5

 m of deflection.

60

(a)

(b)

Figure 33: Second-order ray aberrations for a mirror with minor axis radius

a

= 2 mm and case . (a) Astigmatism with linear dependence on field variables, where atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

. Given the linear nature of the field dependence, one can multiply the results by any value of

u

2

or

v

2

to determine aberrations for larger fields.

(b) Spherical aberration that behaves similar to coma due to a cubic dependence on normalized exit pupil variables .

61

Third-Order Aberrations

The symmetry-allowed, 3 rd

-order aberrations are defined as

.

The aberration coefficients again use the definition . The third-order aberration coefficients with

variable stop position

,

p

, are shown in Equation 23 on the following page.

62

(23)

63

Whereas all 2nd-order coefficients with variable stop go to zero for a normal-incidence mirror, none of the 3rd-order coefficients go to zero for a normal-incidence mirror. If these terms are simplified to the normal incidence case, then they agree with published

Seidel aberration terms [110]. The section Seidel Aberration Terms for Normal Incidence

Mirror below presents the comparison.

For variable stop location with an object at infinity

, the terms reduce as shown in Equation 24 on the following page.

64

(24)

65

For

2

= 0 and the infinite-conjugate case, spherical aberration remains zero (terms 𝑑̃

1𝑝,𝑟

1

→∞

, 𝑑̃

4𝑝,𝑟

1

→∞

, and 𝑑̃

11𝑝,𝑟

1

→∞

).

Setting the

stop at the mirror

plane rids the system of 3 rd

-order distortion terms 𝑑̃

7

, 𝑑̃

10

, 𝑑̃

15

, and 𝑑̃

16

, and the astigmatic term 𝑑̃

13

,

(25)

The 𝑑̃

6

coefficient goes to zero for

2

= 0.

For an object at infinity ( ), we utilize the relationships and

. The normalized

d

-terms for the

object at infinity and stop at the mirror

may be expressed as,

66

(26)

All spherical aberration coefficients, 𝑑̃

1,𝑟

1

→∞

, 𝑑̃

4,𝑟

1

→∞

, and 𝑑̃

11,𝑟

1

→∞

, and the astigmatic coefficient, 𝑑̃

6,𝑟

1

→∞

, go to zero for

2

= 0. Three terms, 𝑑̃

9,𝑟

1

→∞

, 𝑑̃

12,𝑟

1

→∞

and 𝑑̃

14,𝑟

1

→∞ have no dependence on the angle of incidence.

For the object and image at equal distances from the mirror plane ( ), the relationships and hold. We may express the normalized

d

terms

for stop at mirror and r

1

=

r

2

as,

67

(27)

All comatic coefficients, 𝑑̃

2,𝑟

1

=𝑟

2

, 𝑑̃

5,𝑟

1

=𝑟

2

, 𝑑̃

8,𝑟

1

=𝑟

2

, and 𝑑̃

14,𝑟

1

=𝑟

2

, and the astigmatic coefficient, 𝑑̃

6,𝑟

1

=𝑟

2

, go to zero for

2

= 0. For the case, the system proves completely reversible about the stop at the MEMS mirror. For reversible systems, lower orders of coma tend to zero (page 141 in [108]).

Seidel Aberration Terms for Normal-Incidence Mirror

For additional validation of the veracity of the aberration coefficients, the thirdorder-aberration terms may be compared with the more traditional Seidel aberration terms. We evaluate our system at normal incidence by setting , thereby making the system axially symmetric for a circular-boundary mirror. All 2 nd

-order aberration terms

68 evaluate to zero for normal incidence. We set provides

, , and . This

(28) where

S

,

C

,

A

,

F

, and

D

are the spherical, coma, astigmatism, field curvature, and distortion coefficients, respectively.

Equations 4-3 through 4-6 in Mahajan [92] provide axially symmetric aberration coefficients with a shift in stop position for a spherical mirror and Mahajan’s Equation 6-

11 provides an additional wavefront deviation for a conic. Using the case of a paraboloidal mirror (Equation 6-14 in Mahajan [92]), aberration coefficients for a variable stop can be compared. We substitute

L

=

r

2

-

p

and

S

’ =

r

2

. The radius of curvature

R

is negative. For coma, the substitution of

R

into 𝑑̃

2𝑝 and 𝑑̃

8𝑝 results in a coefficient proportional to 𝑥̃𝜌̃

2

. To un-normalize the exit-pupil coordinates (power of 3 in the case of coma), the coefficient is multiplied by

( 𝑎 (𝑟

2 𝑟

2

−𝑝)

)

3

. Using a similar approach for all aberration terms, the terms with variable stop position in Equation (23) agree with

Mahajan (Table 5).

The axially symmetric aberration coefficients (Equation (28)) of a paraboloidal mirror were compared with Mahajan’s Equation 6-65b [92]. This equation is for an object lying at infinity at an angle

from the axis of the mirror. Equation

and the constraints and values provided in Table 6 are used in the 3 rd

-order aberration terms for comparison. Accounting for the difference in signs for the focal length

f

(the analysis in this paper defines it positive and Mahajan's analysis

69 defines it negative for a concave mirror) and converting the

u

2

variable into

*f

, the 3 rd

order terms generate the same aberration equation as Mahajan (Table 7).

Table 5: Comparison of 3 rd

-order, variable-stop-position terms with Mahajan’s axially symmetric terms. All terms agree.

Aberration Coefficient Analysis Values Mahajan's Values

S

C

4

(

2

2 (

2

)

) 4

2

2

2

(

2

(

2

) 4

)

2

( )

4

4 2

2

( )

4 2

A

F

2

2

( )

(

2

) 4

(

2

2

(

2

)

4

)

2

( )

4 2

( )(

4 2

2

)

D

(

2

2 2

(

2

) 4

2

)

2

2

-

( − )

2

( 2

2

)

Table 6: Constraints and values given to variables to compare 3 rd

-order aberration terms with Mahajan’s Equation 6-65b.

Variable

r1 r2 f

1

2 u2 v2 n

Constraint Imposed

positive positive positive

Relationship Imposed

f

-

2

*f

Value Assigned

inf

0

0

1

70

Table 7: Axially symmetric aberration coefficients for an object lying at infinity at an angle

from the axis of a paraboloidal mirror.

Aberration Coefficient

S

C

A

F

D

Values

0

/(4

f

2

)

2

/(2

f

)

0

0

Setting , , and simplifying terms in Equation (12) of order 4 in pupil coordinates, the 3 rd

-order spherical aberration may be written as,

. (29)

The first half of the equation shows that some spherical aberration with dependence only on

p

4

or ( 𝑥̃

2

+ 𝑦̃

2

)

2

exists when not at normal incidence. The second half of the equation has an ( 𝑥̃ 2

+ 𝑦̃ 2

)

𝑥̃ 2

contribution that is 16 times greater than the symmetric portion.

Peak-to-valley 3 rd

-order

Aberration Terms for an Object at Infinity

Figure 34 shows the peak-to-valley 3

rd

-order aberration terms for an

a

= 2 mm mirror with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o where appropriate. Peak-to-valley depends on the shape in the exit pupil. For instance, 𝑑̃

5

has dependence on exit pupil coordinates with relationship 𝑥̃

2 𝑦̃

.

Knowing quadratic and linear dependence in both coordinates, I assume the maximum and minimum values occur at radius of 1 in the exit pupil. This provides 𝑥̃ 2

= 1𝑦̃ 2

.

Substituting into 𝑥̃ 2 𝑦̃

, differentiating with respect to 𝑦̃

, and settting equal to zero indicates that maximums and minimums occur at 𝑦̃

=

±√(1/3). This gives a peak-to-

71 valley of

4

3 √3

. The aberration coefficient also has a multiplier of ½ that is taken into consideration, resulting in peak-to-valley aberration of

2

3 √3 𝑑̃

5

v

2

.

Table 8 summarizes the maximum aberration values at

2

= 0 o

and

2

= 45 o

for the mirror deflected to 30

 m with . One might note that the 3 rd

-order, peak-tovalley values are generally more than an order of magnitude less than 2 nd

-order aberration terms. Both 2 nd

- and 3 rd

-order coma increases with an increase in incidence angle for field variables of 1 o

.

Figure 34: The 3 rd

-order, peak-to-valley aberrations for various deflections and incidence angles of an

a

= 2 mm mirror and . Although atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o for all plots, the vertical axis labels show dependence of the aberration terms on field angles for ease of changing field angles. Plots a, b, c, d, e, f, g, h, i, j, and k are for coefficients 𝑑̃

1

, 𝑑̃

2

, 𝑑̃

3, 𝑑̃

4

, 𝑑̃

5, 𝑑̃

6

, 𝑑̃

8

, 𝑑̃

9

, 𝑑̃

11

, 𝑑̃

12

, and 𝑑̃

14

.

72

Figure 34

– CONTINUED

Figure 34: The 3 rd

-order, peak-to-valley aberrations for various deflections and incidence angles of an

a

= 2 mm mirror and . Although atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o for all plots, the vertical axis labels show dependence of the aberration terms on field angles for ease of changing field angles. Plots a, b, c, d, e, f, g, h, i, j, and k are for coefficients 𝑑̃

1

, 𝑑̃

2

, 𝑑̃

3, 𝑑̃

4

, 𝑑̃

5, 𝑑̃

6

, 𝑑̃

8

, 𝑑̃

9

, 𝑑̃

11

, 𝑑̃

12

, and 𝑑̃

14

.

73

Figure 34

– CONTINUED

Figure 34: The 3 rd

-order, peak-to-valley aberrations for various deflections and incidence angles of an

a

= 2 mm mirror and . Although atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o for all plots, the vertical axis labels show dependence of the aberration terms on field angles for ease of changing field angles. Plots a, b, c, d, e, f, g, h, i, j, and k are for coefficients 𝑑̃ 𝑑̃

2

, 𝑑̃

3, 𝑑̃

4

, 𝑑̃

5, 𝑑̃

6

, 𝑑̃

8

, 𝑑̃

9

, 𝑑̃

11

, 𝑑̃

12

, and 𝑑̃

14

.

1

,

74

Figure 34

– CONTINUED

Figure 34: The 3 rd

-order, peak-to-valley aberrations for various deflections and incidence angles of an

a

= 2 mm mirror and . Although atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o for all plots, the vertical axis labels show dependence of the aberration terms on field angles for ease of changing field angles. Plots a, b, c, d, e, f, g, h, i, j, and k are for coefficients 𝑑̃ 𝑑̃

2

, 𝑑̃

3, 𝑑̃

4

, 𝑑̃

5, 𝑑̃

6

, 𝑑̃

8

, 𝑑̃

9

, 𝑑̃

11

, 𝑑̃

12

, and 𝑑̃

14

.

1

,

75

Figure 34

– CONTINUED

Figure 34: The 3 rd

-order, peak-to-valley aberrations for various deflections and incidence angles of an

a

= 2 mm mirror and . Although atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o for all plots, the vertical axis labels show dependence of the aberration terms on field angles for ease of changing field angles. Plots a, b, c, d, e, f, g, h, i, j, and k are for coefficients 𝑑̃ 𝑑̃

2

, 𝑑̃

3, 𝑑̃

4

, 𝑑̃

5, 𝑑̃

6

, 𝑑̃

8

, 𝑑̃

9

, 𝑑̃

11

, 𝑑̃

12

, and 𝑑̃

14

.

1

,

76

Figure 34

– CONTINUED

Figure 34: The 3 rd

-order, peak-to-valley aberrations for various deflections and incidence angles of an

a

= 2 mm mirror and . Although atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o for all plots, the vertical axis labels show dependence of the aberration terms on field angles for ease of changing field angles. Plots a, b, c, d, e, f, g, h, i, j, and k are for coefficients 𝑑̃ 𝑑̃

2

, 𝑑̃

3, 𝑑̃

4

, 𝑑̃

5, 𝑑̃

6

, 𝑑̃

8

, 𝑑̃

9

, 𝑑̃

11

, 𝑑̃

12

, and 𝑑̃

14

.

1

,

77

Table 8: Peak-to-valley aberration values for the case atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

, and .

a

= 2 mm,

= 30

 m,

Figure 35, Figure 36, Figure 37, Figure 38, and Figure 39 show peak-to-valley,

3 rd

-order aberrations for terms with non-linear dependence on field angles. Figure 35

shows 𝑑̃

9,𝑟

1

→∞

and 𝑑̃

14,𝑟

1

→∞

, which have no dependence on base-ray incidence angle.

These astigmatic terms exceed 50 nm around 2 o

of field angle for deflections greater than

78

15

 m. The other figures show the other astigmatic terms 𝑑̃

3,𝑟

1

→∞

and 𝑑̃

6,𝑟

1

→∞

, for

2

= 0 o

, 5 o

, 10 o

, and 45 o

, respectively. The term 𝑑̃

6,𝑟

1

→∞

is zero for

2

= 0 o

. Although astigmatism resulting from field parallel (

u

1

,

u

2

) to the long axis of the MEMS mirror is greater overall, it shows less change with an increase in angle of the mirror. At

2

= 0 o

,

5 o

, 10 o

, and 45 o

, the astigmatic term 𝑑̃

3,𝑟

1

→∞

exceeds 50 nm around 2 o

of field angle for deflections greater than 15

 m. At

2

= 5 o

and 10 o

, the astigmatic term 𝑑̃

6,𝑟

1

→∞

with quadratic dependence on

v

2

remains below 60 nm for all deflections of the mirror out to

10 o

field angle in the

v

2

-direction. At

2

= 45 o

, however, it becomes significant with only a 2 o

field angle.

79

Figure 35: The 3 rd

-order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror with 𝑑̃

14,𝑟

1

→∞

, have no dependence on

For instance, if atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

. These two aberration terms, 𝑑̃

9,𝑟

1

→∞

, and

2

. The field angle along the diagonal means that

u

2

=

v

2

.

) = 3.54

o

, then field angle along the diagonal is 5 o

.

80

Figure 36: The 3 rd

-order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 0 o

incidence angle with . The term 𝑑̃

6,𝑟

1

→∞

is zero at normal incidence. The field angle in the

u

2 direction is atan(

u

2

/

r

2

).

81

Figure 37: The 3 rd

-order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 5 o

incidence angle with . The field angle in the

u

2 direction is atan(

u

2

/

r

2

). The field angle in the

v

2 direction is atan(

v

2

/

r

2

).

82

Figure 38: The 3 rd

order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 10 o

incidence angle with . The field angle in the

u

2 direction is atan(

u

2

/

r

2

). The field angle in the

v

2 direction is atan(

v

2

/

r

2

).

83

Figure 39: The 3 rd

order, peak-to-valley aberrations for various deflections and field angles of an

a

= 2 mm mirror at 45 o

incidence angle with . The field angle in the

u

2 direction is atan(

u

2

/

r

2

). The field angle in the

v

2 direction is atan(

v

2

/

r

2

).

RMS Wavefront Aberration

The RMS wavefront aberration is defined as

, (30)

84 where

A

is the area of the exit pupil and the integration occurs over the area of the exit pupil. Numerically, this can be calculated as

.

Assuming a full exit pupil, we have less than 3% error when compared with raytracing results in Zemax for the RMS calculations. For all graphs in this chapter,

31,064 sample points over a rectangular grid are used to calculate the wavefront surface.

A mask filters out all points except those within the circular aperture of the exit pupil.

The deflection of the mirror,

, has a step size of 1.6

 m. All graphs show a mirror with deflection ranging from 0

 m to 40

 m unless otherwise specified. As

increases, 1/

f

also increases. The imaging equation provides a relationship of 1/

r

1

+1/

r

2

= 1/

f

. Blue on the graph (< 50 nm) represents

/10 imaging for visible light and

/20 imaging for

= 1

 m. Green (< 100 nm) represents

/10 imaging for



= 1

 m. Infinite-conjugate imaging may be found on the 1/

r

1

= 0 axis.

Figure 40 shows RMS values for

2

= 0 o

, 5 o

, 10 o

, and 45 o

. The left column shows

RMS values for no-field imaging with

u

2

=

v

2

= 0.

The right column shows RMS values for 5 o

of field along the diagonal in the image plane (atan(u

2

/r

2

) = atan(v

2

/r

2

) = -3.54

o

).

The corresponding u

2

and v

2

values are opposite the sign of r

2

for 5 o

of field along the diagonal.

This definition will be used for all RMS plots with diagonal field angles.

It results in the largest angle of incidence for a point object at coordinate

u

2 along the long axis of the MEMs mirror, thus providing the greatest and most limiting RMS value for diagonal field. Quite a range of deflection may be achieved with good performance up to

10 o

incidence angle. However, without any compensation of astigmatism (most likely

85 requiring extra elements to the system), the RMS error proves significant with 5 o

field angle along the diagonal for incidence angles greater than or equal to 5 o

. The bifurcation in the no-field, normal-incidence RMS plot shows the minimization of RMS values when either the object or the image is at infinite-conjugate. In contrast, non-normal-incidence mirrors show a minimization of RMS error along a diagonal where

r

1

=

r

2

.

86

No field 5 o along the diagonal

2

=0 o

0.03

0.02

0.01

400

350

300

250

200

150

100

50

0

0.03

0.02

0.01

400

350

300

250

200

150

100

50

0 0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

400 400

2

=5 o

0.03

0.02

0.01

350

300

250

200

150

100

50

0

0.03

0.02

0.01

350

300

250

200

150

100

50

0 0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

2

=10 o

0.03

0.02

0.01

400

350

300

250

200

150

100

50

0

0.03

0.02

0.01

400

350

300

250

200

150

100

0

-0.1

-0.05

1/r

0

1

(mm

-1

0.05

)

0.1

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

50

0

2

=45 o

0.025

0.02

0.015

0.01

0.005

400

350

300

250

200

150

100

50

0.025

0.02

0.015

0.01

0.005

400

350

300

250

200

150

100

50

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

Figure 40: Wavefront RMS for

 shows no-field imaging with

u

2

2

=

= 0

v

2

0.1

0 0

-0.1

0 o

, 5 o

, 10

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

o

, and 45 o

shown in rows. The left column

= 0 and the right column has 5 o

of field along the diagonal in the image plane. All graphs show the mirror with a deflection ranging from

0

 m to 40

 m.

87

Figure 41 provides a plot of wavefront RMS for 5

o

incidence angle with 3 o

along the diagonal. The comparative plots with 3 o

in only

u

2

- or

v

2

- direction indicate that an increase in field in

u

2

introduces more aberrations than an increase in

v

2

. A field of 1 o along the diagonal, only in the

u

2

direction, and only in the

v

2

direction for



= 45 o

is

shown in Figure 42. The graphs do not differ much from each other due to such a small

angle of field-of-view. For

r

1

= 150 mm and

f

= 70.8 mm, a field of 1 o

along the diagonal, only in the

u

2

-direction, and only in the

v

2

-direction for



= 45 o

results in RMS values of

202.5 nm, 203.4 nm, and 200.0 nm. The RMS value is lesser along the diagonal than for strictly 1 o

in the

u

2

-direction due to the fact that the angle in

u

2

-direction is reduced to

0.707

o

for this case.

88

0.03

0.02

0.01

400

350

300

250

200

150

100

50

0 0

-0.1

2

-0.05

=5 o

, 3 o

0

(mm

-1

)

0.05

0.1

1/r

1

along the diagonal

400 400

0.03

0.02

350

300

250

200

150

0.03

0.02

350

300

250

200

150

0.01

100

50

0.01

100

50

0

-0.1

-0.05

1/r

0

1

(mm

-1

)

0.05

0.1

2

=5 o

, atan(

u

2

/

r

2

)=3 o

, and

v

2

=0

0 0

-0.1

-0.05

0

(mm

-1

)

0.05

0.1

1/r

1

2

=5 o

,

u

2

=0, and atan(

v

2

/

r

2

)=3 o

0

Figure 41: Wavefront RMS for

= 0

 m to 40

 m for 5 o

incidence angle. The top plot shows a field of 3 o

along the diagonal. The bottom two plots show the aberrations with 3 o field in the individual field coordinates.

Figure 43 and Figure 44 include wavefront RMS for mirrors with different radii

for incidence angles 0 o

and 45 o

, respectively. One might note that a smaller-diameter mirror has greater optical power than a larger-diameter mirror with the same deflection.

In practice, smaller mirrors tend to move less far for the same stress and voltage than larger mirrors (Equation (1)). All graphs show



= 0

 m-40

 m.

89

0.025

0.02

0.015

0.01

0.005

400

350

300

250

200

150

100

50

0

0.025

0.02

0.015

0

-0.1

2

-0.05

=45 o

, 1 o

0

(mm

-1

)

0.05

1/r

1

along the diagonal

0.1

400

350

0.025

300

250

200

0.02

0.015

400

350

300

250

200

0.01

150

100

0.01

150

100

0.005

0.005

50 50

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

2

=45 o

, atan(

u

2

/

r

2

)=1 o

, and

v

2

=0

0.1

0 0

-0.1

-0.05

1/r

1

0

(mm

-1

0.05

)

2

=45 o

,

u

2

=0, and atan(

v

2

0.1

/

r

2

)=1 o

0

Figure 42: Wavefront RMS for



= 0

 m to 40

 m for 45 o

incidence angle. The top plot shows a field of 1 o

along the diagonal. The bottom two plots show the aberrations with 1 o field in the individual field coordinates.

Utilizing different fabrication and actuation schemes, a convex shape in addition

to a concave shape can be attained with deformable-membrane mirrors. Figure 45 shows

the wavefront RMS values for

2

= 0 o

, 5 o

, 10 o

, and 45 o

. These graphs have deflection increments of 6.3

 m. The left column shows no-field imaging and the right column has

5 o

of field along the diagonal in the image plane (atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = -3.54

o

) with

u

2

and

v

2

being opposite in sign than

r

2

. All graphs show deflection ranging from -20

 m to 20

 m to easily compare a total 40

 m deflection with the purely concave mirrors

90

analyzed in Figure 40. One might note that

10

 m of deflection yields more aberrations than +10

 m of deflection. The total usable stroke with 5 o

field along the diagonal proves similar for

2

= 5 o

and

2

= 45 o

. At

2

= 0 o

, the concave/convex mirror has better performance over the entire deflection range with 5 o

field along the diagonal. It also has a more useful deflection range for no-field imaging with greater incidence angles.

0.06

0.04

0.02

a

=1.5 mm

0.03

0.02

0.01

a

=2 mm

-0.05

1/r

1

0

(mm

-1

)

0.05

a

=2.5 mm

400

350

300

250

200

150

100

50

0 0

-0.1

0.025

0.02

0.1

400

350

300

0

-0.1

0.015

-0.05

1/r

1

0

(mm

-1

)

0.05

a

=3 mm

0.1

400

350

300

0.015

0.01

250

200

150

0.01

250

200

150

0.005

100

50

0.005

100

50

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

0

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

0

Figure 43: Wavefront RMS for MEMS mirrors with different radii. All graphs show results for

2

= 0 o

with a deflection range of 0

 m to 40

 m. Mirrors with larger radii have less optical power for the same deflection as smaller radii mirrors.

400

350

300

250

200

150

100

50

0

91

0.05

0.04

a

=1.5 mm

400

350

300

250

0.025

0.02

a

=2 mm

400

350

300

250

0.03

0.015

0.02

0.01

200

150

100

50

0.01

0.005

200

150

100

50

0

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

a

=2.5 mm

0.1

0

0

-0.1

-0.05

1/r

1

0

(mm

-1

0.05

)

a

=3 mm

0.1

0

400

350

0.012

400

350

0.015

0.01

0.01

300

250

200

0.008

300

250

200

0.006

0.005

150

100

50

0.004

0.002

150

100

50

0

-0.1

-0.05

0.1

0

0

-0.1

0

1/r

1

0

(mm

-1

)

0.05

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

Figure 44: Wavefront RMS for MEMS mirrors with different radii. All graphs show results for

2

= 45 o

with a deflection range of 0

 m to 40

 m.

92

No field 5 o along the diagonal

2

=0 o

0.01

0

-0.01

200

150

100

50

0

400

350

300

250

0.01

0

-0.01

200

150

100

50

0

400

350

300

250

-0.02

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

-0.02

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

2

=5 o

0.015

0.01

0.005

0

-0.005

-0.01

-0.015

400

350

300

250

200

150

100

50

0

0.015

0.01

0.005

0

-0.005

-0.01

-0.015

400

350

300

250

200

150

100

50

0

-0.1

-0.05

1/r

1

0

(mm

-1

0.05

)

0.1

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

2

=10 o

0.015

0.01

0.005

0

-0.005

-0.01

-0.015

200

150

100

50

0

400

350

300

250

0.015

0.01

0.005

0

-0.005

-0.01

-0.015

200

150

100

50

0

400

350

300

250

-0.1

-0.05

1/r

0

(mm

-1

)

0.05

0.1

-0.1

-0.05

1/r

1

0

(mm

-1

)

0.05

0.1

0.01

0.005

400

350

300

0.01

0.005

400

350

300

0

250

200

0

250

200

2

=45 o

-0.005

150

100

-0.005

150

100

-0.01

-0.01

50 50

-0.1

-0.05

1/r

1

0

(mm

-1

)

Figure 45: Wavefront RMS for

 shows no-field imaging with

u

2

0.05

2

=

= 0

v

2

0.1

0 o

, 5 o

, 10

-0.1

0 o

, and 45

-0.05

1/r

1

0

(mm

-1

0.05

)

0.1

o

shown in rows. The left column

=0 and the right column has 5 o

of field along the diagonal in the image plane. All graphs show the mirror with a deflection ranging from

-20

 m to 20

 m.

93

Comparison of a Circular and Elliptical Mirror at 5 o

Incidence Angle

Sometimes only circular-boundary mirrors may be available to the designer and not elliptical. The designer then has a choice between using a beam splitter and maintaining normal incidence with the circular mirror or using the mirror slightly off-

axis. Table 9 illustrates RMS values for 10 and 20

 m of deflection of an elliptical and circular-boundary mirror used at 5 o

incidence angle. The circular mirror has RMS values of approximately 20 nm and 40 nm for



= 10

 m and



= 20

 m, respectively, with nofield at

2

= 5 o

for both instances. RMS for no-field conditions of an elliptical-boundary mirror proves negligible. The circular-boundary mirror has greater than

/10 (for visible light) RMS values for most field conditions, whereas the elliptical-boundary mirror is less than

/10 for 10

 m of deflection. The circular mirror does show smaller RMS values for field only along its short axis than the elliptical-boundary mirror.

94

Table 9: Comparison of an elliptical and a circular mirror used at 5 o

incidence angle with

r

1

= 200 mm and

a

= 2 mm.

Conditions

RMS (

m) of

at 5 o

2

=5 o

elliptical mirror used

incidence angle

2

=5 o

,

u

1

=

v

1

=0 mm

2

=5 o

,

u

1

=12.3 mm or 5 o

,

v

1

=0 mm

2

=5 o

,

u

1

=0 mm,

v

1

=12.3 mm or 5 o

2

=5 o

,

u

1

=0 mm,

v

1

=24.6 mm or 10 o

0.0001 (0.004)







2

=5 o

,

u

1

=12.3 mm or 5

v

1

=12.3 mm or 5 o

 o

,

0.050 (0.100)

RMS values are for



=10

 m (

=20

 m)

Conditions

RMS (

m) of circular mirror used at 5 o incidence angle

2

=0 o

,

u

1

=12.3 mm,

v

1

=0 mm

2

=0 o

,

u

1

=24.6 mm,

v

1

=0 mm

2

=0 o

,

u

1

=12.3 mm,

v

1

=12.3 mm

2

=0 o

,

u

1

=12.3 mm,

v

1

=24.6 mm

0.019 (0.037)

0.075 (0.150)

0.038 (0.074)

0.094 (0.188)

2

=0 o

,

u

1

=24.6 mm,

v

1

=12.3 mm

RMS values are for



=10

 m (

=20

 m)

0.094 (0.188)

Strehl Ratio

According to Planchon et al. [111], the Strehl ratio may be accurately approximated by

(31) for small aberrations. Typically, a Strehl ratio greater than 0.8 implies essentially diffraction-limited performance. For imaging with



= 633 nm, an RMS value of 45 nm provides

S

= 0.8.

95

Conclusions and Guidelines for a Paraboloidal Dynamic Varifocal Mirror

This section provides guidelines for maximum recommended deflections for given incidence angles. It also offers suggestions for field limitations to maintain

S

= 0.8.

The recommendations are for

r

1

= ∞ and

a

= 2 mm. I first discuss the dominant 2 nd

-order aberrations, which are generally an order of magnitude greater than the 3 rd

-order aberrations. Next, I present guidelines for 3 rd

-order aberrations, including

2

= 0 o imaging. Finally, I briefly review what others in the literature suggest for designing optical systems to address the attendant astigmatism and coma that comes with off-axis imaging.

Comparison of 2 nd

- and 3 rd

-order Terms

The MEMS mirrors described in this dissertation may correct spherical aberration with their concentric electrodes. The non-symmetric astigmatism and coma aberrations

depend on location in field and prove more difficult to correct. Figure 46 shows 2

nd

-order,

3 rd

-order, and combined 2 nd

- and 3 rd

-order astigmatism and coma aberration terms versus

F

for an

2

= 5 o

and

a

= 2 mm mirror with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o at

r

1

= ∞. It also includes the 2 nd

-order, field-independent terms that present similar to coma. The 2 nd

- and

3 rd

-order coma terms include the 2 nd

-order, field-independent terms. An increase in center displacement (or alternatively a decrease in

F

) leads to greater astigmatism and coma.

Observation of the peak-to-valley curves confirms this. An increase in center displacement of the MEMS mirror of 25

 m increases 3 rd

-order, peak-to-valley astigmatism and coma by approximately 6% and 35%, respectively. Astigmatism

96 dominates the aberrations. Coma plays a more significant role for

F

< 100 than greater

F

values. The 2 nd

- and 3 rd

-order coma exceeds 2 nd

order astigmatism for

F

< 11.5. The 3 rd

order coma exceeds 3 rd

-order astigmatism for

F

< 20. The 2 nd

-order, field-independent terms exceed 3 rd

-order astigmatism for F

< 36. Figure 47 is a similar plot, but has larger

aberrations for an

2

= 45 o

and

a

= 2 mm mirror with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

at

r

1

= ∞.

10

3

3rd order coma terms

2nd order field independent terms

2nd and 3rd order coma terms

3rd order astigmatism terms

2nd order astigmatism terms

2nd and 3rd order astigmatism terms

10

2

10

1

10

0

10

-1

0 50 100 150 200

F

250 300 350 400

Figure 46: The 2 nd

-order terms, 3 rd

-order terms, and total peak-to-valley coma and astigmatism decrease with increasing

F

for an

2

= 5 o

and

a

= 2 mm mirror with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o at

r

1

= ∞.

Coma and astigmatism versus angle of incidence are shown in Figure 48 for an

F

=50 system with and

a

= 2 mm for atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

. Astigmatism contributes more of the total aberration for the system as the angle of incidence increases.

The 3 rd

-order coma increases with increasing

2

. The 2 nd

-order, field-independent terms

97 exceed the 3 rd

-order coma terms for

2

greater than 2.8

o

. Overall, astigmatism proves the dominant aberration for these large

F

mirrors. Coma terms influence imaging most in comparison with astigmatism at very small incidence angles. The 3 rd

-order coma increases much more than 3 rd

-order astigmatism with increasing angle of incidence.

10

3

3rd order coma terms

2nd order field independent terms

2nd and 3rd order coma terms

3rd order astigmatism terms

2nd order astigmatism terms

2nd and 3rd order astigmatism terms

10

2

10

1

10

0

10

-1

0 50 100 150 200 250 300 350 400

Figure 47: The 2 nd

-order terms, 3 rd

F

-order terms, and total peak-to-valley coma and astigmatism decrease with increasing

F

for an

2

= 45 o

and

a

= 2 mm mirror with atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o at

r

1

= ∞.

Third-order aberrations dominate 2 nd

-order aberrations for small incidence angles

(Figure 49). The figure is for

r

1

= ∞ and is valid for all center displacements of a parabolic mirror. The larger the field-of-view in the image plane, the more 3 rd

-order terms influence image quality. RMS values for a mirror at



= 10

 m are provided for reference. RMS values for only 2 nd

-order astigmatism versus only 2 nd

-order, fieldindependent aberrations were determined, but are not shown. The results indicated that

98 for most deflections and angles of incidence, 2 nd

-order astigmatism RMS values prove greater than 2 nd

-order, field-independent RMS values for field angles along the diagonal less than 0.1

o

. The field-independent term does grow more greatly with increasing displacement of the mirror than the astigmatism terms though. While 2 nd

-order aberrations limit off-axis mirrors,

2

= 0 o

mirrors do not have any 2 nd

-order aberrations.

Their largest sources of aberrations are coma terms, 𝑑̃

2,𝑟

1

→∞

and 𝑑̃

12,𝑟

1

→∞

, and astigmatism terms, 𝑑̃

3,𝑟

1

→∞

, 𝑑̃

9,𝑟

1

→∞

, and 𝑑̃

14,𝑟

1

→∞

. These coma and astigmatism terms also negatively impact off-axis mirrors. Furthermore, 𝑑̃

1,𝑟

1

→∞

, 𝑑̃

5,𝑟

1

→∞

, and 𝑑̃

6,𝑟

1

→∞ significantly contribute to the spherical, coma, and astigmatism of off-axis parabolic

varifocal mirrors. The reader should reference Table 8 for more details.

Figure 48: Peak-to-valley coma and astigmatism versus angle of incidence,

2

, for an

F

= 50 system with and

a

= 2 mm for atan(

u

2

/

r

2

) = atan(

v

2

/

r

2

) = 1 o

.

99

Figure 49: Field angle along the diagonal (where image plane variables have the opposite sign as

r

2

) in the image plane versus incidence angle for which RMS of either 2

3 rd

-order aberrations dominate for

r

1 have a larger RMS than 2 nd

= ∞. The field angle at which 3 rd

-order aberrations

-order aberrations remains the same for all deflections,



of the mirror. The total RMS for a mirror at 10

 m deflection is also shown. nd

-order or

Incidence Angle and Field Limitations

The 2 nd

-order aberrations compose the biggest limitation to imaging performance of these mirrors, as was shown in the previous section. This section provides guidelines for maintaining

S

= 0.8 for



= 633 nm (RMS < 45 nm). We use RMS values generated by the entire -function for

r

1

= ∞ and

a

= 2 mm.

Table 10 provides incidence angles that maintain

S

= 0.8 for a given center displacement of a mirror with no field. For the mirrors presented in this dissertation

(

≤ 19

 m), they should be able to be used at

2

= 45 o

without significant degradation

100 of image quality. One would expect a parabolic mirror at

= 40

 m to have good imaging performance up to

2

= 7 o

.

Table 10: Maximum recommended angle of incidence for given deflections to maintain

S

= 0.8 for

= 633 nm,

r

1

= ∞, and

a

= 2 mm.

Table 11 and Table 12 provide maximum field guidelines to maintain appropriate

Strehl ratios for

r

1

= ∞. A mirror at

2

= 5 o

and with

= 10

 m should not have a field angle that exceeds atan(

u

2

/

r

2

) = 2.8

o

with no field in

v

2

-direction. A mirror at

2

= 5 o

and with

= 10

 m should not have a field angle that exceeds atan(

v

2

/

r

2

) = 3.4

o

with no field in

v

2

direction.

Table 11: Maximum recommended field angle of atan(

u

2

/

r

2

) with deflections to maintain

S

= 0.8 for

= 633 nm,

r

1

= ∞, and

a

= 2 mm.

v

2

= 0 for given

101

Table 12: Maximum recommended field angle atan(

v

2

/

r

2

) with deflections to maintain

S

= 0.8 for

= 633 nm,

r

1

= ∞, and

a

= 2 mm.

u

2

= 0 for given

Table 13 provides maximum field angle along the diagonal in the image plane

(

u

2

=

v

2

) to maintain appropriate Strehl ratios for

= 633 nm,

r

1

= ∞, and

a

= 2 mm. A mirror at

2

= 5 o

and with

= 10

 m should not have a field angle along the diagonal that exceeds 2.9

o

. A mirror at

2

= 0 o

can have 3 o

of field for up to

= 30

 m. For design purposes, the maximum incidence angle with

S

=0.8 image quality for a field angle of 5 o along the diagonal and

= 10

 m is calculated as

2

= 0.9

o

.

Table 13: Maximum recommended field angle along the diagonal (

u

2

=

v

2

) for given deflections to maintain

S

= 0.8 for

= 633 nm,

r

1

= ∞ and

a

= 2 mm.

Zemax Overview

This section uses Zemax to show spot diagrams for 2 nd

-order dominant aberrations and 3 rd

-order dominant aberrations.

102

One Wave of 2 nd

-order Astigmatism for Object at Infinity. To appreciate the influence of 2 nd

-order aberrations, we can calculate the maximum values

u

2

and

v

2

for peak-to-valley astigmatism to remain under 1

for the infinite-conjugate case. The 2 nd

order astigmatic expressions and provide

(32)

As an example, one would expect approximately 1

of astigmatism for

u

2

= 2 mm and

v

2

= 2 mm when using a 5 mm x 7.071 mm elliptical-boundary mirror

(

Z

= .0016

X

2

+.0032

Y

2

) at 45 o

incidence angle with

= 1

 m and

= 20

 m (Table 14).

As shown in Figure 49, 2 nd

-order aberrations dominate RMS values for this case. The distance

r

1

is infinity and

r

2

is 110 mm. Figure 50, Figure 51, and Figure 52 show optical

path difference fans, wavefront surfaces, and through-focus spot diagrams, respectively.

These plots show full wavefront aberration, but are dominated by second-order terms.

Approximately 1

of astigmatism is observed for (

u

2

= 2 mm,

v

2

= 0 mm) and

(

u

2

= 0 mm,

v

2

= 2 mm). The peaks and valleys of the surface shape in the exit pupil for

(

y

2

-

x

2

) and

xy

rotate 45 o

. Since these are linearly dependent on

u

2

and

v

2

, the 45 o

rotation between (

u

2

= 2 mm,

v

2

= 0 mm) and (

u

2

= 0 mm,

v

2

= 2 mm) may be observed in the wavefront surfaces and through-focus spot diagram. After passing through the Gaussian focal plane, the spot diagram rotates 90 o

. This corresponds with the two focal lines associated with rays originating from the

x-

and

y

-plane in the exit pupil. While

103 dominated by astigmatism, the OPD fans are also influenced by field-independent, 2 nd

order aberration.

Table 14: Constraints and values given to variables to observe 1

of 2 nd

-order astigmatism.

Variable Relationship Imposed Value Assigned

Z lens r

1

inf

r

2



1

2

u

2

v

2

a

-

2

20

-45 o m

0 or 2 mm

0 or 2 mm

2.5 mm

b

0 mm, 0 mm 2 mm, 0 mm

0 mm, 2 mm 2 mm, 2 mm

Figure 50: Zemax optical path difference fans for four different image locations (

u

2

,v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

(

5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is

110 mm.

Peak to Valley: 0.2

0 mm, 0 mm

104

y

Peak to Valley: 1.1

2 mm, 0 mm

x

Peak to Valley: 1.1

Peak to Valley: 1.5

0 mm, 2 mm 2 mm, 2 mm

Figure 51: Zemax wavefront surfaces without tilt for four different image locations (

u

2

, v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial

(

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is

110 mm.

0 mm, 0 mm

2 mm, 0 mm

0 mm, 2 mm

2 mm, 2 mm

Figure 52: Zemax through-focus spot diagrams for four different image locations (

u

2

, v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial

(

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is

110 mm.

105

Figure 53 and Figure 54 show 1

of astigmatism generated by an image at a field height of 22 mm (or equivalently 11.2

o

) for a circular-boundary, normal-incidence mirror with the same optical power as the previous 45 o

-incidence mirror (Table 15).

The through-focus spot diagram (Figure 55) does not show evidence of field-

independent, 2 nd

-order aberration. Because there are no field-independent, 2 nd

-order aberrations, the OPD fans also appear symmetric.

Table 15: Constraints and values given to variables to observe 1

of 2 nd

-order astigmatism.

Variable Relationship Imposed Value Assigned

Z lens r

1

inf

r

2



1

2

u

2

v

2

a

-

2

14

0

 o m

0 or 22 mm

0 or 22 mm

2.5 mm

b

106

0 mm, 0 mm

22 mm, 0 mm

0 mm, 22 mm

Maximum Scale: 1

22 mm, 22 mm

Figure 53: Zemax optical path difference fans for a 5 mm circular-boundary mirror represented as an even polynomial (

Z

= .00224

X

2

+.00224

Y

2

) with

= 1

 m, where

r

1 and

r

2

are infinity and 112 mm, respectively. (b) Wavefront surfaces. (c) Through-focus spot diagrams at the four different image locations.

y

Peak to Valley: 0



 scale e-5

)

0 mm, 0 mm

Peak to Valley: 1

22 mm, 0 mm

x

Peak to Valley: 1

Peak to Valley: 1.9

0 mm, 22 mm 22 mm, 22 mm

Figure 54: Zemax wavefront surfaces without tilt for a 5 mm circular-boundary mirror represented as an even polynomial (

Z

= .00224

X

2

+.00224

Y

2

) with

= 1

 m. Variable

r

1 is infinity and

r

2

is 112 mm.

107

0 mm, 0 mm

22 mm, 0 mm

0 mm, 22 mm

22 mm, 22 mm

Figure 55: Zemax through-focus spot diagrams for a 5 mm circular-boundary mirror represented as an even polynomial (

Z

= .00224

X

2

+.00224

Y

2

) with

= 1

 m. Variable

r

1 is infinity and

r

2

is 112 mm.

Finally, a mirror at small incidence angle may represent a compromise that avoids a beam splitter without suffering unduly from the 2 nd

-order aberration terms. For 5 o

field angle, we previously indicated that less than 1 o

incidence angle should maintain

S

= 0.8 for

a

= 2 mm at

r

1

= ∞. For comparison purposes to the two previous Zemax cases, we investigate a 5 o

incidence angle mirror with

a

= 5 mm. Figure 56 and Figure 57 show 1

 of astigmatism generated by an image at a field height of 22.3 mm (or equivalently

11.4

o

). The mirror has the same optical power as the previous 0 o

and 45 o

incidence

mirrors (Table 16). The OPD fans and wavefront surfaces indicate some asymmetry,

which is most likely due to the small amount of 2 nd

-order, field-independent aberration.

108

The through-focus spot diagram (Figure 58) does not show significant evidence of field-

independent, 2 nd

-order aberration.

Table 16: Constraints and values given to variables to observe 1

of 2 nd

-order astigmatism.

Variable

Z lens r

1

Relationship Imposed Value Assigned

inf

r

2



1

2

u

2

v

2

a

-

2

14.2

5 o

 m

0 or 22.3 mm

0 or 22.3 mm

2.5 mm

b

109

0 mm, 0 mm 22 mm, 0 mm

0 mm, 22 mm

Maximum Scale: ±2

22 mm, 22 mm

Figure 56: Zemax optical path difference fans for four different image locations (

u

2

, v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

5.000 mm x 5.019 mm elliptical-boundary mirror represented as an even polynomial

(

Z

= .00226

X

2

+.00227

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is

110 mm.

y

Peak to Valley: 0.02



0 mm, 0 mm

Peak to Valley: 0.62

22 mm, 0 mm

x

Peak to Valley: 1.4

Peak to Valley: 1.8

0 mm, 22 mm 22 mm, 22 mm

Figure 57: Zemax wavefront surfaces without tilt for four different image locations (

u

2

, v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

(

5.000 mm x 5.019 mm elliptical-boundary mirror represented as an even polynomial

Z

= .00226

X

2

+.00227

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is

110 mm.

110

0 mm, 0 mm

22 mm, 0 mm

0 mm, 22 mm

22 mm, 22 mm

Figure 58: Zemax through-focus spot diagrams for four different image locations (

u

2

, v

2

) to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A

5.000 mm x 5.019 mm elliptical-boundary mirror represented as an even polynomial

(

Z

= .00226

X

2

+.00227

Y

2

) is analyzed here with

= 1

 m. Variable

r

1 is infinity and

r

2

is

110 mm.

One Wave of 2 nd

-Order Astigmatism for Object and Image at Equal Distances. To compare aberrations for

r

1

= ∞ versus a reversible system with

r

1

=

r

2

(image and object planes are symmetric about the MEMS mirror), we calculate the maximum values

u

2

and

v

2

for astigmatism to remain under 1

for the

r

1

=

r

2

case. The 2 nd

-order astigmatic expressions and provide

(33)

111

Based on these calculations, one would expect approximately 1

of astigmatism for

u

2

= 4 mm and

v

2

= 4 mm when using a 5 mm x 7.071 mm elliptical-boundary mirror

(

Z

= .0016

X

2

+.0032

Y

2

) at 45 o

incidence angle with

= 1

 m and

= 20

 m. This provides the same field angle as the case of 1

for an object at infinity with the same size mirror and angle of incidence. The distances

r

1 and

r

2

are both 220.97 mm. Figure 59,

Figure 60, and Figure 61 show optical path difference fans, wavefront surfaces, and

through-focus spot diagrams. Approximately 1

of astigmatism is observed for

u

2

= 4 mm,

v

2

= 0 mm and

u

2

= 0 mm,

v

2

= 4 mm. As indicated by 𝑐̃

1

and 𝑐̃

5

going to zero for this case, 2 nd

-order, field-independent aberration does not exist.

Table 17: Constraints and values given to variables to observe 1

of 2 nd

-order astigmatism.

Variable Relationship Imposed Value Assigned

Z lens r

1

221 mm

r

2



1

2

u

2

v

2

a

-

2

20

-45 o m

0 or 4 mm

0 or 4 mm

2.5 mm

b

112

0 mm, 0 mm

4 mm, 0 mm

0 mm, 4 mm

Maximum Scale: 1

4 mm, 4 mm

Figure 59: Zemax optical path difference fans to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 7.071 mm ellipticalboundary mirror represented as an even polynomial (

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variables

r

1 and

r

2

are 221 mm.

y

Peak to Valley: 0

0 mm, 0 mm

Peak to Valley: 1

4 mm, 0 mm

x

Peak to Valley: 1

Peak to Valley: 1.4

0 mm, 4 mm 4 mm, 4 mm

Figure 60: Zemax wavefront surfaces to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 7.071 mm elliptical-boundary mirror represented as an even polynomial (

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m.

Variables

r

1 and

r

2

are 221 mm.

113

0 mm, 0 mm

4 mm, 0 mm

0 mm, 4 mm

4 mm, 4 mm

Figure 61: Zemax through-focus spot diagrams to verify approximately 1 wave of astigmatism due to 2 nd

-order aberration terms. A 5.000 mm x 7.071 mm ellipticalboundary mirror represented as an even polynomial (

Z

= .0016

X

2

+.0032

Y

2

) is analyzed here with

= 1

 m. Variables

r

1 and

r

2

are 221 mm.

114

MEMS Mirror Represented as an Even Polynomial with 2 nd

- and 4 th

-order Coefficients

For this section the aberration coefficients are determined for a mirror with deflection defined as . The parabasal coefficients are

(34)

Setting these equal to zero, the two conditions, , must be upheld to rid the system of focus error and parabasal astigmatism. Substituting

for

C

a2

and assuming no 4 th

-order contribution (i.e.,

C

a4

= 0), these two relationships are the same as for the parabolic mirror case analyzed above. The 2 nd

-order coefficients also show agreement with the parabolic case and do not have any dependence on

C

a4

,

(35)

As one would expect, symmetric 4 th

-order control over the mirror shape with

C

a4

, only affects the 3 rd

-order spherical aberration coefficients. They are 𝑑̃

1

, 𝑑̃

4

, and 𝑑̃

11

. The

3 rd

-order coefficients are defined in Equation 36 on the following page.

115

(36)

116

CHAPTER 5

EXPERIMENTAL VERIFICATION

Optical Test Layout

For this chapter, we test the SOI mirrors described in Chapter 2 to verify the

aberration coefficients and -function in Chapter 4. Figure 62 shows the exit pupil plane

of a test mirror at 45 o

incidence angle. It also shows the variables and local coordinate systems used in the aberration analyses. A 1x relay images the exit pupil plane onto a

Thorlabs Shack-Hartmann Wavefront Sensor WFS150-5C. We vary previously defined object distance,

r

1

, and object positions,

u

1

and

v

1

(Figure 63) for testing.

v

1

u

1

x

X

MEMS

2

=45 o

Exit Pupil

Plane

r

1

Y, y r

2

u

2

v

2

Figure 62: A point source at

u

1

=

v

1

= 0 in the object plane illuminates a MEMS mirror at

45 o

incidence angle. The image plane has coordinates of

u

2

,

v

2

. The exit pupil plane (

x

,

y

coordinates) is normal to the image plane.

Laser

SM fiber

117

Mirror

Plane

MEMS

2

=45 o

Exit Pupil

Plane

r

1

1x Relay

v

1

u

1

r

1

x

X

MEMS

2

=45 o

Exit Pupil

Plane

Y, y

1x Relay

WFS

(a)

Conjugate Image

Plane

y

WFS

x

(b)

Figure 63: (a) A point source (



= 633 nm) illuminates the MEMS mirror from a distance,

r

1

. A 1x relay consisting of two

f

= 30 mm achromats images the MEMS mirror onto the wavefront sensor. When the natural focus of the MEMS mirror equals

r

1

, the MEMS mirror collimates the green light rays as shown. The exit pupil plane is imaged onto the

WFS. (b) The same optical set-up as shown in (a) with local coordinate systems shown.

Results

For the measured experimental wavefronts, the wavefront sensor fits 24 Zernike terms to the measured wavefront. These 24 terms (neglecting piston Z1, tip Z2, tilt Z3, and defocus Z5 as described in Appendix C) are used to reconstruct the wavefront of the specified experiment. The results shown under Experimental Results also subtract the baseline wavefront measured with zero volts applied to the mirror from each measured wavefront to account for error arising from the relay. These experimental wavefronts are compared with wavefronts generated from the entire -function. Chapter 4 separates 2 nd and 3 rd

-order aberration terms out of the Taylor series expansion of the -function.

Wavefronts generated from these 2 nd

- and 3 rd

-order terms are also shown. As one may observe, up to 3 rd

-order terms do accurately model the aberrations of these mirrors.

Results for a 45 o

and 5 o

incidence angle mirror are shown below.

118

Process-1 4.000 mm x

5.657 mm 45 o

Incidence Angle Mirror

The Characterization section in Chapter 2 presents surface profiles for the process-1 4.000 mm x 5.657 mm 45 o

incidence angle mirror (Figure 17, Figure 19, Figure

21, and Figure 22). Although the optical surface is 4.000 mm x 5.657 mm, lateral under-

cutting extends the membrane beyond the optical surface to 4.340 mm x 5.997 mm. The dimensions of the membrane dominate its behavior and the elliptical boundary prescribes a 43.6

o

angle of incidence. We give the mirror a label of process-1 4 mm 45 o

mirror in reference to its optical surface. A deflection of 5

 m on the mirror corresponds with a voltage profile of E1 = 195 V, E2 = 173 V, E3 = 194 V, and E4 = 206 V. A deflection of

10

 m on the mirror corresponds with a voltage profile of E1 = 234.5 V, E2 = 198 V,

E3 = 255 V, and E4 = 285 V. Figure 64, Figure 65, and Figure 66 show the experimental

results with different amounts of field and deflection of the mirror at

r

1

= 284 mm,

r

1

= 141 mm, and

r

1

= 76 mm, respectively. Generally, the experimental results show agreement with overall shape of aberrations, but they typically are slightly more in

magnitude. Figure 67 shows how minimal the aberrations should be for no-field imaging,

but no experiments were done for this case.

119

2 nd and 3 rd order aberration coefficients

-2 -2 -2

a

2

=43.6

o ,

= mm,



=10

 m,

u

1

=15.6 mm,

v

1

=8.7 mm,,

r

1

=284 mm

-1

0

1

-1 0 x (mm)

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

-1 0 x (mm)

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

2

-2

-2

2

2

-2

-2

2

2

-2 -1 0 x (mm)

S+baseline

1 2

-2

a

= mm,



2

=43.6

=5

 o

, m,

u

1

=15.6 mm,

v

1

r

1

=8.7 mm,

=284 mm

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

2

-2

-2

-1 0 x (mm)

1 2

2

-2

-2

-1 0 x (mm)

1 2

2

-2

-2

-1 0 x (mm)

S-baseline

1 2

2

=43.6

o

,

a

= mm,



=10

 m,

u

1

=6.9 mm,

v

1

=0 mm,

r

1

=284 mm

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0 (

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

2

-2

-2

-1 0 x (mm)

1 2

2

-2

-2

-1 0 x (mm)

1 2

2

-2

-2

-1 0 x (mm)

S-baseline

1 2

2

=43.6

o

,

a

= mm,



=5

 m,

u

1

=6.9 mm,

v

1

=0 mm,

r

1

=284 mm

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

2

-2 -1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

Figure 64: Wavefronts generated from the entire -function, 2 nd

- and 3 rd

-order aberration terms, and Zernike terms from the WFS for

r

1

= 284 mm.

120

a

2

=43.6

= o

, mm,



=10

 m,

u

1

=12.1 mm,

v

1

=8.7 mm,

r

1

=141 mm

a

2

=43.6

o

,

= mm,



=5

 m,

u

1

=12.1 mm,

v

1

=8.7 mm,

r

1

=141 mm

a

2

=43.6

o

,

= mm,



=10

 m,

u

1

=3.4 mm,

v

1

=0 mm,

r

1

=141 mm

a

2

=43.6

o ,

=



=5 mm,

 m,

u

1

=3.4 mm,

v

1

=0 mm,

r

1

=141 mm

-2

-1

0

1

2

-2

-2

-1

0

1

2

-2

-2

-1

0

1

2

-2

-2

-1

0

1

2

-2

-1 0 x (mm)

1

-1 0 x (mm)

1

-1 0 x (mm)

1

-1 0 x (mm)

1

2

2

2

2

1

0.5

0

(

 m)

-0.5

-1

-1.5

1

0.5

0

(

 m)

-0.5

-1

-1.5

1

0.5

0

(

 m)

-0.5

-1

-1.5

1

0.5

0

(

 m)

-0.5

-1

-1.5

-2

-1

0

1

2

-2

-2

-1

0

1

2

-2

-2

-1

0

1

2

-2

-2

-1

0

1

2

-2

2 nd and 3 rd order aberration coefficients

-1 0 x (mm)

1 2

1

0.5

0

(

 m)

Experimental results

SL40A 4mm45degNo2 234p5 198 255 285V zern

-2

1

-1

0.5

0

0 (

 m)

-0.5

-0.5

-1

1

-1

-1.5

-1.5

2

-2 -1 0 1 2

-2

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1 0 x (mm)

1 2

2

-2

-2

-1 0 x (mm)

S-baseline

1 2

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1 0 x (mm)

1 2

2

-2 -1 0 1 2

-2

-0.5

-1

-1.5

1

0.5

0

(

 m)

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

Figure 65: Wavefronts generated from the entire -function, 2 nd

and 3 rd

order aberration terms, and Zernike terms from the WFS for

r

1

= 141 mm.

121

-2

2 nd and 3 rd order aberration coefficients

-2

Experimental results

-2

2

=43.6

o

,

a

= mm,



=10

 m,

u

1

=10.6 mm,

v

1

=8.7122 mm,

r

1

=76 mm

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

2

-2

-2

-1 0 x (mm)

1 2

2

-2

-2

-1 0 x (mm)

1 2

2

-2

-2

-1 0 x (mm)

S-baseline

1 2

a

2

=43.6

o ,

= mm,



=10

 m,

u

1

=1.9 mm,

v

1

=0 mm,

r

1

=76 mm

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

2

-2 -1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

Figure 66: Wavefronts generated from the entire -function, 2 nd

- and 3 rd

-order aberration terms, and Zernike terms from the WFS for

r

1

= 76 mm.

a

2

=43.6

=



=10

 o mm,

, m,

u

1

=0 mm,

v

1

=0 mm,

r

1

=284 mm

2

=43.6

a

=



=10

 o mm,

, m,

u

1

v

1

=0 mm,

=0 mm,

r

1

=141 mm

2

=43.6

a

=



=10

 o mm,

, m,

u

1

=0 mm,

r v

1

1

=0 mm,

=76 mm

-2 -2 -2

-1

0

1

1

0.5

0 (

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0

(

 m)

-0.5

-1

-1.5

-1

0

1

1

0.5

0 (

 m)

-0.5

-1

-1.5

2

-2 -1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

2

-2 -1 0 x (mm)

1 2

Figure 67: Wavefronts generated from the entire -function to demonstrate theoretical aberrations for no-field imaging for

r

1

= 284 mm,

r

1

= 141 mm, and

r

1

= 76 mm.

122

Process-1 4.000 mm x

4.015 mm 5 o

Incidence Angle Mirror

The Characterization section in Chapter 2 presents surface profiles for the process-1 4 mm 5 o

mirror (Figure 18 and Figure 20). Lateral under-cutting extends the

membrane beyond the optical surface to 4.450 mm x 4.465 mm. The dimensions of the membrane dominate its behavior and the elliptical boundary prescribes a 4.7

o

angle of

incidence. Figure 68 shows the experimental results with different amounts of field and

displacement of the MEMS mirror at

r

1

= 284 mm. Generally, the experimental results do not show a significant amount of aberration. The experimental results do appear to have more spherical aberration than the theoretical predictions. These results are for equipotential voltages of 200 V (

= 5

 m) and 247 V (

= 10.9

 m). The shape of the

MEMS mirror for these experiments did not minimize spherical aberrations. However, subtracting out the spherical aberration contribution predicted from phase shift interferometry did not result in better agreement with theoretical wavefront shape predictions.

123

-2

0.5

-2

2 nd and 3 rd order aberration coefficients

0.5

-2

Experimental results for spherical minimized

0.5

2

=4.7

o

,

a

= mm,



=10.9

 m,

u

1

v

1

= -19.1 mm,

=17.6 mm,

r

1

=284 mm

-1

0

1

2

-2

-2

-1 0 x (mm)

1

a

2

=4.7

o ,

=



=5 mm,

 m,

u

1

v

1

= -19.1 mm,

=17.6 mm,

r

1

=284 mm

-1

0

1

2

-2

-2

-1 0 x (mm)

1

0 (

 m)

2

-0.5

0.5

0

(

 m)

-1

0

2

-2

-2

-1

0

1

-1 0 x (mm)

1

0

(

 m)

-1

0

1

2

-0.5

0.5

0

(

 m)

-1

0

2

-2

-2

-1 0 x (mm)

1

0 (

 m)

2

-0.5

0.5

0

(

 m)

2

-0.5

0.5

1

2

-2

-2

-1 0 x (mm)

1 2

-0.5

0.5

1

2

-2

-2

-1 0 x (mm)

1 2

-0.5

0.5

2

=4.7

o

,

a

= mm,



=10.9

 m,

u

1

= -1.5 mm,

v

1

=0 mm,

r

1

=284 mm

-1

0

1

2

-2

-2

-1 0 x (mm)

1

0

(

 m)

-1

0

1

0

(

 m)

-1

0

1

0

(

 m)

2

-0.5

0.5

2

-2

-2

-1 0 x (mm)

1 2

-0.5

0.5

2

-2

-2

-1 0 x (mm)

1 2

-0.5

0.5

2

=4.7

o

,

a

= mm,



=5

 m,

u

1

= -1.5 mm,

v

1

=0 mm,

r

1

=284 mm

-1

0

1

2

-2

0 (

 m)

-1

0

1

0

(

 m)

-1

0

1

0

(

 m)

-1 0 x (mm)

1 2

-0.5

2

-2 -1 0 x (mm)

1 2

-0.5

2

-2 -1 0 x (mm)

1 2

-0.5

Figure 68: Wavefronts generated from the entire -function, 2 nd

- and 3 rd

-order aberration terms, and Zernike terms from the WFS for

r

1

= 284 mm. Note: The shape of the MEMS o mirror was not minimized to reduce spherical aberration, as was done for the 45 incidence mirror results above.

124

CHAPTER 6

DEMONSTRATION OF FOCUS CONTROL AND SPHERICAL ABERRATION

CORRECTION FOR MULTI-LAYER DISCS

This chapter outlines the membrane mirror optical requirements in terms of stroke and aberration compensation required for multi-layer focusing for current digital versatile discs (DVD) and blu-ray disc (BD) standards. I demonstrate an adjustable range of at least 1.6

 m peak wavefront spherical aberration correction at a membrane displacement of 7

 m, which should be sufficient capability for quadruple-layer BDXL

TM

discs. The majority of this chapter comes from a conference proceedings paper [70]. The demonstration is done with a 3 zone, single silicon wafer mirror at 45 o

incidence angle.

Optical discs present a cost-effective means for portable data storage. Of these discs, BDs provide the greatest amount of storage due to their high NA of 0.85 and short,

405 nm, wavelength light, resulting in a diffraction-limited spot size with diameter less

than one-half of DVDs’ spot size (Table 18)

[123]. Commercially, the BDXL

TM

format has increased the capacity of such discs to 100 gigabytes (GB) of read-write storage on triple-layer discs and 128 GB of write-once storage on quadruple-layer discs. Read-only octalayer [124] and 20-layer discs with 500 GB of storage [125] have been proposed for the future. These multi-layer discs not only need accurate and fast focus control, but the associated variation in optical path length through the glass disc medium comes with greater spherical aberration. The induced aberrations of the objective lens when not used at its ideal infinite-conjugate configuration may also prove significant at 0.85 NA.

Current optical pick-up heads utilize liquid crystals [126], magnification change of the

125 objective lens [127], diffractive optical elements [128], diffractive-refractive elements

[129], hologram optical elements [130, 131], a wavelength-selective filter [132] or deformable mirrors [32-34, 80, 133] for spherical aberration compensation. The advantages for the deformable mirrors are that they are achromatic and typically have fast response times, where speed is important for reading and writing data quickly. Switching between layers also requires focus control. As a more compact alternative to the translation of lenses with motors, some investigators have proposed deformable mirrors for focus control [32, 53, 134]. Aoki et. al. demonstrated a single-actuator ellipticalboundary mirror at 45 o

incidence angle for compact focus control and spherical aberration correction of a dual-layer BD [34, 133]. This design is novel in that it alters the stress distribution of the device to control its shape as it deflects under piezoelectric actuation. I present here the use of a large-stroke 3 mm x 4.24 mm elliptical deformablemembrane mirror at 45 o

incidence to achieve a very compact optical system capable of fast multi-layer focusing in a DVD or BD optical pick-up unit (OPU). The MEMS mirror replaces a lens translation mechanism and liquid crystal compensator, resulting in a single optical element to control both focus depth and compensation of attendant focusdependent spherical aberration.

Table 18: Optical disc format specifications.

CD

DVD

Blu-ray

NA Wavelength

0.45 780 nm

0.6

0.85

650 nm

405 nm

Cover layer thickness

1.2 mm

600

 m

100

 m

126

MEMS Requirements for Focusing and Aberration

Correction of Multi-layer DVD and BDXL

TM

Discs

A 3 mm x 4.24 mm elliptical-boundary mirror with an approximate 25

 m air gap used at 45 o

incidence in the back focal plane of a disc-read objective lens (Figure 4 and

Figure 6) provides both focus control and spherical aberration correction to compensate

change in the glass thickness for multi-layer discs. Table 19 specifies requirements for a

quadruple-layer BDXL disc. The major and minor axes of the elliptical mirror and its electrodes are in proportion of

, so that they project with circular symmetry along the optical axis. I assume the objective lens is well corrected for spherical aberrations for infinite-conjugate imaging with the focus occurring at the deepest layer in the disc (

L0

for

BDXL

TM

). To address other layers in the disc, the MEMS mirror is deflected, resulting in a shorter focal distance within the disc (

L1-L3

for BDXL

TM

). For dual-layer DVD, the full-range focus shift

Z

between the two layers is 25

 m, while quadruple-layer

BDXL

TM

has four layers with aggregate change in thickness

Z

of 46.5

m (Figure 4). I

calculate the necessary change in MEMS focus and corresponding nominal deflection

based on paraxial analysis. Table 19 tabulates the results.

Table 19: Focus control and spherical aberration correction requirements for multi-layer

DVDs and BDs.

Maximum change in disc thickness

Objective lens focal length

MEMS focal length

f m

Deflection

of MEMS required

Peak spherical aberration due to change in disc thickness

Dual-layer

DVD disc

Quadruple-layer

BDXL

TM

disc

25

 m

46.5

 m

2.33 mm

1.76 mm

340 mm

106 mm

2.4

 m

7.5

 m

0.16

 m, 0.26



1.3

 m, 3.3



127

The spherical aberration due to refraction at the air-disc interface is calculated by assuming a point object is located at a depth

Z

in the disc medium, with its Gaussian image point located in air at a distance

Z/n

from the air/disc interface. I assume the system is well corrected for the deepest layer with Z = Z

1

, and calculate the residual spherical aberration when the system is refocused at a different depth

Z

2

where

is the distance between layers. This wavefront aberration

W

is given by,

, (37) where is the ray angle in glass, and the maximum wavefront aberration corresponds to

. The peak spherical aberration due to the maximum change in disc thickness for DVDs and the quadruple-layer BDXL

TM

discs are also shown in Table 19.

Demonstration of Necessary Performance

I demonstrated MEMS focus and aberration compensation with a Panasonic

EYLGUDM128 DVD objective lens (NA = 0.6) using a HeNe laser with

= 633 nm, because a blu-ray objective lens was not readily available. Since the four-layer

BDXL

TM

disc has the most stringent requirements for the membrane lens, I present a demonstration with the DVD objective lens that is representative of focusing and aberration correction in a BDXL

TM

disc. Specifically, I construct the experiment so that the overall membrane peak deflection is 7.5

 m. This corresponds to the range of focus control necessary for quadruple-layer BDXL

TM

format. I adjust the glass layer thickness so that the peak wavefront spherical aberration is 1.3

 m, corresponding to

128 the peak spherical aberration incurred when changing focus in BDXL

TM

from the deepest layer to the shallowest.

Using the interferometer set-up shown in Figure 69, I took interferometric

images of the objective lens aperture. The 633 nm light is spatially filtered using single-mode fiber and expanded to achieve uniform illumination of the aperture. The two lenses with

f

= 150 mm and

f

= 200 mm form an image of the DVD objective lens aperture on the CCD camera, with a magnification of 1.33. Initially, I placed a glass layer of 620

 m total thickness on top of a flat optical mirror behind the DVD objective lens to represent the cover layer of a DVD disk for which this lens is well compensated. The glass layer is composed of a stack of microscope cover slips (BK7,

n

= 1.52), using index matching oil between each slide. Reference interferograms are recorded, verifying that the lens is indeed diffraction limited when focused through this thickness of glass.

Next, the deformable mirror is deflected to approximately 7.5

 m and the thickness of the cover slips is reduced to 370

 m, with the mirror/glass assembly repositioned to the location of best focus. This glass thickness change (250

 m differential) causes 1.6

 m of spherical aberration at NA = 0.6, similar to what is expected for BDXL

TM

(NA = 0.85) when changing the focus from the deepest to most shallow layer (47

 m differential). That spherical aberration has a peak value of

1.3

 m, which is 3.3 waves of aberration at λ = 405nm. I chose a value with slightly higher spherical aberration due to availability of cover slips with only specific

129 thicknesses. Interferograms are recorded for a variety of control voltage combinations until the aberration is minimized.

DVD Objective Lens is imaged.

CCD

3

2

1

HeNe

Laser

Beam

Coupling

Lens

SM Fiber

Beam

Collimating

Lens f = 200 mm f = 150 mm

45 o

MEMS Mirror

DVD Objective Lens

620

 m Thick Glass

Mirror

Reference

Mirror

Figure 69: Interferometer for observing aberrations at the DVD objective lens.

Finally, an 835 nm Michelson interferometer was used to directly characterize the MEMS mirror. The interferometer provides surface shape data for the mirror with the specified control voltages. The relationship between the mirror displacement at

normal incidence and the single-pass optical path length in Figure 69 for testing of the

mirror with the DVD objective lens is

Results

Figure 70 shows representative interferograms. The DVD lens is well corrected

for the cover layer thickness of 620

m of glass. Figure 70a and Figure 70b verify that the initially flat MEMS mirror introduces no aberration into the system. Figure 70c shows

130 the aberrated interferogram when the glass thickness is reduced by 250

 m to a total thickness of 370

 m. The peak spherical aberration introduced measures at 3.2

 m in double-pass reflection. The single-pass value is therefore in agreement with the 1.6

 m peak aberration predicted for this glass thickness using Equation (37).

With the MEMS mirror deflected to 6.7

 m at mirror center, with 200 V on all electrodes, we note that the peak round-trip spherical aberration is reduced by

approximately one fringe (Figure 70 (d)), showing that the shape of the mirror with

uniform voltage on all three electrodes includes some spherical aberration that is partially compensating the system aberration. We empirically adjusted the voltages until the

fringes were straight (Figure 70 (e)). The required voltages were 189 V, 202 V, and

225 V on electrodes 1, 2, and 3, respectively. Based on fringe analysis, we find the resultant residual single-pass spherical aberration of the system to be less than 130 nm peak-to-valley. The difference between the single-pass spherical aberration of the system

with the optical flat (Figure 70c) and with the MEMS mirror (Figure 70e) is

approximately 1.6

 m.

Figure 71 shows 6 th

-order curve fits for the membrane shape corresponding to the two different electrode voltage combinations used in the imaging demonstration. This data is taken along the short axis of the membrane and is plotted versus a normalized

 pupil radius.

131

620

 m glass

Optical flat

(a)

MEMs mirror

(b)

370

 m glass

Optical flat

(c)

370

 m glass

Electrode 1: 200 V

Electrode 2: 200 V

Electrode 3: 200 V

(d)

MEMs mirror

Electrode 1: 189 V

Electrode 2: 202 V

Electrode 3: 225 V

(e)

Figure 70: (a) Interferogram with an optical flat in place of the MEMS mirror and cover glass with thickness of 620

 m. (b) Interferogram with un-deflected MEMS mirror and

620

 m cover glass; both (a) and (b) show nearly flat fringes indicating the absence of spherical aberration. (c) Interferogram with 370

 m thick cover glass shows approximately 3.2

 m of spherical aberration in double-pass reflection, corresponding to

1.6

 m single-pass. (d) Interferogram with 200 V on all three electrodes, peak mirror deflection is 6.7

 m and round trip aberration is reduced by approximately 1 fringe. (e)

Three-zone control minimized single-pass spherical aberration to less than 130 nm peakto-valley.

132

The quadratic coefficient of the polynomial curve fit describes a pure defocus of the system, while coefficients above 2 nd

order in the pupil coordinate indicate the amount of spherical aberration the mirrors introduce to the system. Using the polynomial

coefficients determined in Figure 71, the 4

th

- and 6 th

-order terms (corresponding to

primary and secondary spherical aberration) are plotted in Figure 72 for the two different membrane shapes. Figure 72a shows just the 4

th

- and 6 th

-order terms, while Figure 72b

shows the same data plotted with balancing defocus to better illustrate the deviation of the membrane shape from a purely parabolic profile.

We observed asymmetric aberration with the MEMS mirror at 45 o

incidence.

Figure 73 shows the system aberration with all tilt and defocus removed, with the best

compensation of spherical aberration along the vertical aspect. Residual optical path difference along the horizontal aspect of the pupil is still present, which is a consequence of the loss of radial symmetry introduced by the off-axis deformable mirror. For the demonstration here, with approximately 7

 m membrane deflection, we observe less than

 one fringe in double-pass reflection, implying that the single pass aberration is less than

316 nm peak-to-valley.

133

0

0

-1

Surface Profile

6th Order Curve Fit

-1

Surface Profile

6th Order Curve Fit

-2

-2

-3

-3

-4

-4

-5

-5

-6

-6

-7

-1 -0.5

0 0.5

Normalized Pupil Radius

1

-7

-1 -0.5

0 0.5

Normalized Pupil Radius

1

Electrode 1: 200 V

Electrode 2: 200 V

Electrode 1: 189 V

Electrode 2: 202 V

Electrode 3: 200 V Electrode 3: 225 V

(a) (b)

Figure 71: The 6 th

-order, even-polynomial fits ( ) to the measured displacements of the membrane with (a) 200 V equipotential, a center displacement of

6.7

 m, and

a

2

= 7.72,

a

4

= -1.85 and

a

6

= 0.74 and (b) 189 V, 202 V, and 225 V on electrodes 1, 2, and 3, respectively, a center displacement of 7

 m, and

a

2

= 7.50,

a

4

= -1.80 and

a

6

= 1.23.

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2

-1.4

0

134

0

-0.2

0.25

0.2

0.15

-0.4

-0.6

0.1

0.05

-0.8

-1 0

-0.05

-1.2

-1.4

0

Electrode 1: 200 V, Electrode 2: 200 V, Electrode 3: 200 V

Electrode 1: 189 V, Electrode 2: 202 V, Electrode 3: 225 V

0.2

0.4

0.6

Normalized Pupil Radius

(a)

0.8

Electrode 1: 200 V, Electrode 2: 200 V, Electrode 3: 200 V

Electrode 1: 189 V, Electrode 2: 202 V, Electrode 3: 225 V

1

-0.1

0

0.25

0.2

0.15

0.1

0.05

0

Electrode 1: 200 V, Electrode 2: 200 V, Electrode 3: 200 V

Electrode 1: 189 V, Electrode 2: 202 V, Electrode 3: 225 V

0.2

0.4

0.6

Normalized Pupil Radius

(a)

-0.05

0.8

1

-0.1

0 0.2

0.4

0.6

Normalized Pupil Radius

0.8

1

(b)

Figure 72: Spherical aberration introduced by the surface shape of the membrane including (a) plots of only and (b) the same data with balancing defocus added.

Electrode 1: 200 V, Electrode 2: 200 V, Electrode 3: 200 V

Electrode 1: 189 V, Electrode 2: 202 V, Electrode 3: 225 V

0.2

0.4

0.6

Normalized Pupil Radius

(b)

0.8

1

Figure 73: Asymmetric aberration evident when defocus and tilt are removed, and spherical aberration is balanced along the vertical aspect of the pupil (the short axis of the membrane).

135

Chapter Summary

The three-zone deformable mirror exhibits adjustable spherical aberration, introducing optical path variation with peak magnitude in excess of 1.6

 m superimposed on a defocus optical path difference of 10

 m (7

 m membrane deflection). This range of adjustment should be sufficient to correct for the 1.3

 m spherical aberration introduced by the variable cover layer thickness of quadruple-layer BDXL

TM

discs. Furthermore, the range of defocus the membrane mirror can provide is sufficient to address all four layers of the BDXL

TM

format.

The complete design of a BDXL

TM read head with integral MEMS deformable mirror would require attention to several other details. The deformable mirror causes illumination of the lens with a converging beam of light, while the lens was designed for a collimated illumination beam. This will introduce additional aberrations that must be considered. Since we did not have detailed information about the aspheric shape of the

DVD objective lens used in our experiments, we simulated the aberrations of a similar

0.6 NA lens illuminated with a converging beam, and we observe that the spherical aberration introduced by the lens partially compensates the spherical aberration introduced by the change in cover layer thickness. In this case, the MEMS membrane mirror must balance the residual aberration of the system. This residual aberration in a typical BDXL

TM system may be less than the 1.3

 m optical path difference we calculated in Table 2, but may require compensation of higher-order terms. At NA = 0.6, we observe that the best surface shape already contains significant coefficient values up to 6 th

order in the pupil coordinate (as observed in Figure 71b). It is possible that yet

136 higher order correction may be necessary at NA = 0.85. A mirror to accomplish this balance may benefit from four or more electrode zones.

The residual asymmetric aberration we observe due to 45 o

off-axis illumination of the membrane lens will represent an upper limit to the useful focus control range of this device. If larger focus shifts are necessary for future blu-ray formats, then the membrane lens would require an electrode pattern that allows for correction of aberrations that lack

axial symmetry (such as the residual aberration shown in Figure 73). This could be

addressed, for instance, with a more complex electrode structure.

We showed spherical aberration correction with a 3 mm x 4.24 mm deformablemembrane mirror, while having a large center displacement. Additionally, electrostatic actuation only requires connections at the electrodes for shaping of the membrane, thus allowing for easy implementation in miniature optical systems. Therefore, we conclude that the elliptical MEMS deformable-membrane lens used at 45 o

incidence angle possesses the inherent characteristics necessary to realize an ultra-compact multi-layer read-write head with integral focus control and aberration correction.

137

CHAPTER 7

AGILE SCANNING USING A MEMS FOCUS CONTROL

MIRROR IN A COMMERCIAL CONFOCAL MICROSCOPE

This chapter comes from a conference proceedings paper [115]. The introduction discusses other investigators’ approaches to achieving fast focus control and/or spherical aberration compensation in confocal microscopy. Although this demonstration shows utility of the MEMS mirror in a bench top microscope, our technology is unique in that it can be easily miniaturized to address endoscopic microscope needs. Again, spherical aberration and focus control over a large optical power range can be achieved by one of our devices instead of needing several components.

Confocal microscopy has proven useful for real-time imaging of tissues, potentially as a replacement to biopsies for the detection of dysplastic or cancerous cells, and routinely for the study of intact animal or plant cells. Traditionally, acquisition of

xy-z

stacks (Figure 8a in Chapter 1) provides information that can be further processed for

observation of arbitrary planes within the stack. Commercial microscopes have begun to

make oblique plane imaging (Figure 8b) and other spatial data acquisition possible. Smith

et al. [135, 136] introduced an additional objective lens and mirror to a two-photon microscope to perform aberration-free oblique plane imaging and has also analyzed resolution capabilities with oblique plane imaging. The ability to directly acquire image data along biologically or clinically relevant cross-sections within the tissue improves temporal resolution, limits photobleaching and/or phototoxicity, and likely improves the usefulness of the data. For instance, one might want to follow a prescribed

z

-trajectory

138

(Figure 8c) within a sample (e.g., follow the curve of a cell membrane). G

ö bel and

Helmchen [137] used a sinusoidal signal on a piezo actuator to drive an objective lens in a two-photon microscope to take in vivo images of dynamic signaling in neuronal dendrites along a prescribed trajectory. Mansoor et al. [138] also demonstrated a magnetically driven objective lens for focus control in confocal microscopy.

Typically, translation of the microscope stage or the objective lens allows for

z

axis scanning in confocal microscopes. This mechanical translation of a massive object proves slow and can introduce vibrations during imaging. Not using an objective lens at its designed imaging configuration and changing the depth of focus in a sample introduces spherical aberration to an imaging system. Investigators have analyzed the effects of specimen structure and change in imaging depth on aberrations, and proposed methods for correcting aberrations [3, 82, 139-142]. Botcherby and Smith demonstrated diffraction-limited imaging over large focal ranges by the use of more than one objective lens in multiphoton microscopy [8, 136], where Botcherby has shown imaging at kHz rates of neuronal activity [143]. Micro-electro-mechanical systems (MEMS) mirrors prove a fast and easily miniaturized alternative method for focus control and aberration compensation. Albert and Sherman demonstrated a deformable MEMS mirror as an adaptive optics element for aberration correction in multiphoton confocal microscopy [7,

144]. Poland et al. [145] performed focus locking to minimize in vivo movement effects with Okotech’s deformable-membrane mirror. Kner et al. [149] performed focus control and aberration correction with Imagine-Optic’s Mirao52D mirror in wide-field microscopy.

139

This chapter presents a novel process-1 3-mm x 3.011-mm MEMS mirror with

four concentric electrodes (Figure 74) for a simple electronic interface that allows focus

control over 123

 m at 0.5-0.64 NA. Since the elliptical-boundary mirror lies at 5 o incidence angle, it does not require a beam splitter. Previously, we showed a mirror at 45 o incidence angle for a compact optical set-up [77] and observed off-axis aberrations [70].

A smaller incidence angle should introduce less off-axis aberration during large deflections of the mirror. This chapter explores the influence of altering voltage potentials on the four electrodes on surface shape and on the axial point spread function of the microscope with the MEMS mirror.

E4

E3

E2

E1

Figure 74: (left) Tilted interferogram showing the flatness (less than 90 nm peak-peak deviation) of the process-1 3-mm x 3.011-mm mirror. (right) Top view with labels on the four electrodes.

Optical Design

We examine a 3 mm x 3.011 mm elliptical-boundary mirror used at 5 o

incidence

near the back focal plane of an Olympus 20x water objective lens (Figure 75). Ensuring a

circular projection of the MEMS mirror on the reflected beam eliminates primary

astigmatism. All elements shown in the light green box in Figure 75 represent the MEMS

module that we add to the Olympus Fluoview 300. The f = -166 mm lens pushes the focal

plane out (Figure 6); initially it is 40

 m deeper than the natural focal plane of the

140 objective lens. The MEMS mirror then can pull it back to 83

 m closer than the natural focal plane, for a total range of 123

 m axial focus for the experiments presented here.

The MEMS mirror lies close to the galvo scan plane, but not exactly at it. This causes the beam to fill a variable percentage of the area in the back aperture of the objective lens with focus adjustment. This equivalently causes a change from 0.50-0.64 in NA as the focus adjusts from shallow to deep.

Figure 75: Olympus Fluoview 300 with MEMS module shown in the green box. The

MEMS module consists of an f = -166 mm lens to push the focal plane in the specimen out from the objective when the MEMS mirror is flat.

141

Olympus

20x

0.5-0.64

NA

Negative

Lens

MEMS

Deflected

Coverslip

Natural focus of objective lens

83

 m

MEMS

0

 m

-40

 m n=1.33

Figure 76: The natural focal plane of the Olympus water objective is defined as 0

 m.

With the MEMS module (including the negative lens) inserted with an un-deflected

MEMS mirror the focus lies at -40

 m. An increase in voltage on the MEMS mirror increases its displacement until the focal plane behind the objective lens lies at 83

 m.

Optical Characterization

Using the Zernike polynomials and mapping used by Fricker in his MATLAB routine ZERNFUN2.m [122], we characterized surface profiles of the mirror with a phase-shift interferometer at wavelength



= 835 nm. Table 20 provides definitions of the three axially symmetric aberrations, which concentric electrodes can influence.

Figure 77 shows three different voltage profiles on the electrodes of the mirror to

attain (a) parabolic shape, (b) positive primary and negative secondary spherical aberration, and (c) negative primary and positive secondary spherical aberration.

142

Term #

5

13

25

(a)

(b)

n,m

for

2,0

4,0

6,0

400

350

350

350

420

420

330

260

Polynomial

50

2

r

2

-1

100

6

r

4

-6

r

2

+1

20

r

-30

r

4

+12

r

2

-1

200

50 100 150 200 raw height image p-p displ = 4611 nm

100

100

200

150

50 100 150 200 raw height image

200

100

200

150

200

50 100 150 200

3000

p-p amplitude

4000

3000

2

2000

2000

1000

1.5

1000

4000

3000

0

2000

1000

-2000

-4000

4000

3000

1000

-2000

-4000

2

0

-1000

0

3000

10

Aberration type

Defocus

Primary spherical

Secondary spherical

20 30 40

2000

P-V = 405 nm; RMS = 31 nm

2000

1000

1000

50

10x

0

100

0

-1000

3000

0

Primary

10 20 30 spherical

Zernike mode number

-200

40 spherical

35 5 15 20 25 30

2000

P-V = 507 nm; RMS = 108 nm

2000

1000

300

200

0

100

-2000

200

0

5

0

10 20 30 -100

Zernike mode number

15 20 25 30 35

40

300

300

330

center displ = -3965 nm

50

0

-1000

P-V = 493 nm; RMS = 89 nm

2000

1000

50

100

0

(c)

420 100

-2000

-100

150

-3000

-1000

150

-200

200

50 100 150 200

-2000

200

5 15 20 25 30

-300

35

Figure 77: Three voltage profiles with corresponding surface shapes and Zernike terms.

The amplitude of each Zernike mode higher than mode 5 (defocus) is shown plotted 10x the measured amplitude for all three spectra. Mode 13 is primary spherical and mode 25 is secondary spherical. (a) A voltage of 350 V on the three inner electrodes and 400 V on the outermost electrode results in a parabolic shape (introducing almost no spherical aberration). (b) Much higher voltage on the outer rings than the centermost electrode yields 230 nm of positive primary spherical aberration and -70 nm of secondary spherical aberration. (c) Having the greatest electrostatic pressure in the center of the membrane results in a more cone-like surface as evidenced by a smaller area of blue (greater than -3000 nm) in the surface profile. It produces -180 nm primary and 75 nm secondary spherical aberration.

Prior to investigating the influence of the focus control mirror on the microscope, we inserted water, a 170

 m cover slip, more water, and a flat optical mirror behind the

143 objective lens and used a depth scan with 0.1

 m axial steps on the microscope to obtain the baseline axial point spread function PSF of the microscope. The intensity of the laser beam, as observed on a power meter, varied by as much as 10% during one PSF

measurement. Figure 78 shows the axial PSF with



= 633 nm of the microscope without any alteration. The strong secondary lobe suggests the system may have some

uncorrected spherical aberration (Figure 78).

2500

2000

1500

1000

500

0

-20 -10 0 10

Axial Displacement (

 m)

20

Figure 78: Axial point spread function (



= 633 nm) of the microscope with no MEMS module inserted.

Figure 79 shows the axial PSFs obtained with the MEMS mirror controlling

different focus depths, with the same voltage applied to all electrodes (equipotential results). We used 0.4

 m axial steps for these measurements. Beginning with 200 V

(at -25

 m axial focus displacement), we applied 300 V, 330 V, 360 V, and 400 V to reach 83

 m. The full width at half maximum (FWHM) at 2

 m axial displacement is

144 measured to be 4

 m. The theoretical FWHM at 0.6 NA with our pinhole diameter of 0.5 aperture unit is 2.8

 m ( [150]).

4500

4000

3500

3000

2500

2000

1500

1000

0.64 NA

4

 m

0.5 NA

500

0

-50 0 50 100

Axial Displacement (

 m)

Figure 79: Axial point spread functions (



= 633 nm) of the microscope with the MEMS module inserted and equipotential voltages applied. The MEMS mirror begins nearly flat for the PSF at an axial displacement of -25

 m. As the MEMS mirror increases power, the axial displacement increases in the positive direction.

Table 21 lists electrode voltage configurations that improved the axial PSFs at

given axial displacements. Figure 80 shows the improved PSFs at their respective axial

displacements with the same coloring as in Figure 79. The PSFs narrowed, showed a

decrease in side lobe structures, and showed an increase in intensity at more positive axial displacements with shaping of the mirror surface.

145

4500

4000

3500

3000

2500

2000

1500

1000

500

0

-50 0 50 100

Axial Displacement (

 m)

Figure 80: Altering voltages on the four electrodes improves the axial PSFs (



= 633 nm)

while maintaining similar focal planes as in Figure 79.

Table 21: Varied voltages on electrodes defined in Figure 74. These voltage

configurations improved the axial PSFs (Figure 80) when compared with equipotential

voltages on the electrodes.

Axial Displacement (

m) E1 (V) E2 (V) E3 (V) E4 (V)

-25 210 200 200 190

2

26

50

83

300

320

355

390

300

330

355

400

300

330

360

410

300

350

370

430

Figure 81 illustrates how altering the voltages on the four electrodes can change

the shape of the axial PSF. Greater potentials on the outer electrodes yield a very poor

PSF (Figure 81a). A high voltage on only the centermost electrode produces an improved

PSF (Figure 81b). Applying 300 V on the inner two electrodes and 400 V on the outer

two electrodes gives a recognizable PSF with a substantially larger intensity on the center

lobe (Figure 81c). All three graphs have the same horizontal and vertical scales.

146

(a)

420

420

330

260

300

300

300

350

1400

1200

1000

800

600

400

200

0

-3

(b)

1400

7 17 27

1200

1000

Axial Displacement (

m)

800

600

400

200

0

0 10 20

37

30

Axial Displacement (

m)

(c)

400

400

300

300

1650

1450

1250

1050

850

650

450

250

0 10 20 30

Axial Displacement ( m)

Figure 81: Axial point spread functions (



= 488 nm) of the confocal microscope for three different voltage profiles. Defocus remained constant for the three measurements.

All horizontal (axial displacement) and vertical (intensity) axes are the same scale. (a)

The point spread function has several large side lobes with the center electrode’s voltage being 160 V less than the outer two electrodes. (b) The side lobes shift to the left-hand side of the main lobe with 350 V on the centermost electrode and 300 V on the outer electrodes. (c) With 300 V on the inner two and 400 V on the outer two electrodes point spread function gains intensity on the central lobe and it narrows to approximately 5

 m

FWHM.

Imaging Results

We compared the performance of electronic focus with the MEMS mirror (i.e., applying a voltage to change its radius of curvature) versus translating the microscope stage in

z

-direction by acquiring images of

Drosophila

third-instar larvae dyed with

mCherry in reflectance mode at 633 nm (Figure 82). Both mechanisms had similar image

quality throughout the focal range. Instead of the 5 o

incidence angle used for the majority

147

of the results in this chapter, Figure 83

shows a similar experiment with the MEMS mirror at 45 o

incidence angle [77] for comparison.

MEMS Focus Control

100

 m

-40

m -25

m 2

m

Z-stage Translation of Microscope Stage

26

m 50

m 83

m

Deeper in tissue

Figure 82: Comparison of focus control by changing electrostatic pressure on a MEMS mirror at 5 o

incidence angle (top) versus translating the microscope while leaving the

MEMS mirror flat (bottom). The image quality appears comparable for both mechanisms.

0

m 10

Change in Focus Depth

m 20

m 30

m 40

m

Microscope

Z-stage

Translation

MEMS

Focus

Control

Figure 83: Images of a Stage 28 chick embryo spinal cord, comparing focusing using zstage translation to focusing with a MEMS mirror at 45 o

incidence angle.

We synchronized the MEMS mirror with the fast scan (

x

-axis) galvanometer

mirror of the Fluoview 300 to perform oblique plane imaging. Figure 84a shows an

oblique scan with the mirror pulling focus from -40

 m to 83

 m over 9 ms during each

148 horizontal scan. The oblique plane lies at 16 o

in the sample. Halving the ramp time doubles the angle of the oblique plane to 32 o

(Figure 84b), while traversing the sample a lesser distance in the

x

-direction. It may be noted that different features can be seen along the two distinct oblique sections through the tissue.

y

111 Hz Drive Frequency

x

222 Hz Drive Frequency

z y x

9 ms

100

 m

100

 m

4.5 ms

-40

m

16 o

83

m

32 o

16 o

direction occurs over 9 ms. (b) An oblique scan occurring in 4.5 ms or at 32

(a) (b)

Figure 84: (a) Oblique scan starting at

z

= -40

 m, increasing to

z

= 83

 m, and returning quickly to

z

= -40

 m to begin the next line at a new

y

-coordinate. The change in

z

o

angle in the sample. Different features present in the first 100

 m of the

x

-coordinate than at a 16 o angle oblique scan due to a different profile in

z

.

Figure 85 illustrates how shaping the mirror surface by altering the voltages can

improve contrast and brightness in an image. With 400 V applied on all electrodes, the axial PSF at 543 nm is poor. With 300 V on the centermost electrode, 400 V on the middle two, and 500 V on the outer electrode, the axial PSF and clarity of a fluorescent image of part of a honey bee stinger (dyed with Invitrogen’s Dil paste) improves.

3500

3000

2500

2000

2000

1500

1000

500

3500

3000

2500

1500

1000

500

0

-40 -20 0 20 40

Axial Displacement (

 m)

80

149

100

 m 100

 m

0

-40 -20 0 20 40

Axial Displacement (

 m)

60

E2=E3=400 V

E4=500 V

100

 m

100

 m

Figure 85: Comparative axial point spread functions (same scaling) and single

x-y

plane fluorescent images of a honey bee stinger, all taken at 543 nm. The voltages correspond with a focal point in the sample at 83

 m in

z

-direction.

In summary, we demonstrated the use of a MEMS mirror for agile focus control over a 123

 m range at 0.5-0.64 NA in a commercial confocal microscope. We synchronized the mirror with the fast scan axis of the microscope to attain oblique plane imaging, where changing the speed of the focus control mirror alters the angle of imaging in the sample. The MEMS mirror did not lie precisely in the back focal plane of the objective lens. Ensuring this would preserve the NA of the objective lens over the entire focal range.

This chapter shows the utility of four concentric electrodes for changing the spherical aberration balance on the mirror surface and the shapes of the axial PSFs of the microscope. In the future, optimization of the voltage profiles on the mirror may further

150 improve performance in the confocal microscope. This could be done in conjunction with a wavefront sensor or by optimizing image metrics such as overall intensity.



S mirrors provide electronic focus control. Their fast response times and small size makes them appealing for agile focus control in both table top and endoscopic imaging systems. Fast imaging along a specified

z

-trajectory should minimize photobleaching in samples, allowing for better longer longitudinal studies, and/or allowing for imaging of time-critical phenomena.

151

CHAPTER 8

DISCUSSION

The focal range capability of deformable MEMS mirrors has greatly improved over the last couple of decades. This dissertation presents a new and simple method that utilizes a one SOI wafer for fabrication of mirrors that continue to push focal range capability. These mirrors are not only capable of large deflections (approximately 22

 m for devices that are approximately 5 mm in diameter), but they are of high optical quality and have high yield. A motivating application for varifocal mirrors is in vivo microscopy, which allows discernment of benign and malignant tissue, making it unnecessary to remove tissue for biopsy. For confocal microscopy in particular, penetration depths rarely exceed 200

 m, so being able to control focus throughout that full range is a reasonable objective for varifocal mirrors. The MEMs mirrors fabricated previous to the dry-release mirrors presented in this dissertation tended to have a maximum of 5

 m deflections. For

0.8 NA imaging in water this leads to a change in focus of approximately 42

 m. With a deflection of 20

 m, this increases to a focal range of 165

 m, nearly a four-fold improvement. For real-time imaging this would provide imaging to much greater depths and provide focus control throughout most of the range available in confocal microscopy.

The single-SOI-wafer fabrication process has room for improvements. The isotropic nature of xenon difluoride causes under-cutting in the lateral direction. This means that the perimeter of the membrane does not match the perimeter of the reflective mirror surface. Lateral etch stops can be used in the future by etching side trenches and

152 filling the trenches with oxide. The bandwidth of the devices is limited by air damping, so 3dB frequencies occur below natural resonant frequencies. Careful design of the vias that allows for sufficient airflow while also having a large enough diffraction angle due to periodic spacing for the diffracted rays to be filtered out could mitigate this problem.

Furthermore, the vias could be larger in size to increase airflow and decrease total etch times, but total light throughput should be concurrently considered.

Current literature on aberrations of reflective mirrors does not capture the dynamic nature of varifocal mirrors. The aberration analysis presented here presents the aberrations in terms of varying focal power and incidence angle of elliptical-boundary mirrors. The veracity of the analysis is shown by experimental testing of the novel

MEMS mirrors presented in this dissertation with a WFS and comparison of the

-function with Zemax results and published aberration coefficients. People in the field now have basic guidelines moving forward for use of these mirrors in optical systems.

These mirrors exhibit the best imaging performance over a narrow field of view, regardless of incidence angle. At 45 o

incidence angle, a parabolic mirror with up to

20

 m deflection, such as the mirrors presented in this dissertation, should have nearly diffraction-limited performance with a narrow field of view.

The aberration analysis distinguishes the different types of ray aberrations and separates the orders up to order three of the ray aberrations that are inherent with these mirrors. Generally, lower order terms degrade image quality more significantly than higher order terms. At normal incidence angle, 2 nd

-order terms are zero. As incidence angle increases, the 2 nd

-order terms more significantly degrade image quality. Non-

153 symmetric aberrations, such as astigmatism and coma, affect image quality more considerably at non-normal incidence angles. They cannot be corrected with the concentric electrodes that we currently use. Astigmatism proves to be the dominant nonsymmetric aberration for these low

F

-number mirrors. Another key difference between using these mirrors at normal incidence angles and non-normal incidence angles is that the RMS plots show normal incidence angle parabolic mirrors perform best at infinite conjugate imaging. If the mirror shape deviates to another conic, then a different result for object and image locations would be optimal. However, off-axis mirrors with concentric electrodes will always prefer symmetry or equal object and image locations (a reversible system), because the electrodes cannot cause an asymmetric shape (like an offaxis paraboloid).

Guidelines in terms of incidence angle and field of view for achieving nearly diffraction-limited performance (

S

> 0.8) are presented. Many useful optical systems do not need this high of a Strehl ratio, and the RMS plots may be used to determine guidelines for lower Strehl ratios. Generally microscopes and some telescopes require a

Strehl ratio of greater than 0.8, but few other imaging instruments. For instance, very high quality cameras can have modulation transfer function curves that are considerably worse than diffraction limited. Additionally, the wide-field microscope discussed in

Chapter 1 used a 45 o

incidence angle mirror with conditions leading to less than 0.8

Strehl ratio according to our analysis. The 0.4 NA 40x microscope images taken with a

±1.7

o

x ±1.3

o

field of view still show discernable 1-2

m tendrils in a fungal mat (Figure

86).

154

1-2

m

Figure 86: Wide field (±1.7

45 o o

x ±1.3

o

) images taken with a 0.4 NA 40x microscope with a

incidence angle mirror that should have less than a 0.8 Strehl ratio.

The aberration analysis assumes the short axis of the elliptical mirror surface is proportional to the long axis times the cosine of the incidence angle to eliminate parabasal astigmatism. One may extend this analysis to tip/tilt mirrors that are also capable of defocus by accounting for the fact the mirror surface stays one size for different incidence angles. This means that incidence angle in the analysis should remain constant, while field angle changes with tip or tilt.

Peak-to-valley of 3 rd

-order spherical aberration terms remain below 100 nm for all of the cases examined. Chapters 2, 6, and 7 show our current devices can compensate

100s of nm of 3 rd

-order spherical aberration. The ability of the MEMS mirror to compensate spherical aberration proves advantageous, because spherical aberration dynamically changes with a change in defocus.

If a MEMs mirror introduces too much aberration to an optical system, then correction of attendant coma and astigmatism may be required. Chang et al. [146] specifically address mitigation of linear astigmatism by the use of two reflective mirrors that have equal and opposite astigmatism. Rogers [147] discusses the use of 2 or 3

155 mirrors, or 2 mirrors and a weak cylindrical lens to mitigate these aberrations. Most likely a similar approach would work to minimize aberrations by use of two active MEMS mirrors.

Not only are coma and astigmatism asymmetric aberrations, but they depend on field position. This limits what can be done to address these aberrations with a single mirror. Any scanned system that addresses a single point in the field of view at one time could adapt on the fly. If the system changes between a limited number of points, then the mirror shape could be specific to those points. For general access to the full field of view, a more general electrode pattern would be required. If more than one dynamic mirror is used in a system, then it may be possible to intentionally link them together to control them in a favorable way to balance the aberrations. Some investigators have moved to a

“woofer tweeter” approach, where one dynamic MEMS mirror performs gross focus control and a second high-order (high electrode count) mirror dynamically compensates aberrations.

Chapter 6 shows imaging performance of a 45 o

incidence angle mirror for compact focus control and spherical aberration correction for optical discs. The results suggest that for 7.5

 m deflection, the mirror performed sufficiently well to image all layers of a quadlayer Blu-ray disc. The residual 2 nd

-order, field-independent aberration was observed. Chapter 7 shows imaging results when synchronizing a mirror with the fast-scan axis of a commercial confocal microscope. It also shows improvement of axial point spread functions at different defocus locations by correction of spherical aberration with four concentric electrodes. In the future, agile focus control over specified

156 trajectories should lead to real-time imaging of fast physical phenomena and should decrease photobleaching and phototoxicity for longitudinal studies. In our experience, dynamic focus control with a deformable membrane mirror reduces complexity of the optical instrument, offering a path for miniaturization of a system that preserves the ability to change the focus throughout the useful field of view.

The large focal range of the novel MEMS mirrors presented in this dissertation combined with the predictions of the aberration analysis show these mirrors have great promise for agile imaging in the future. This could benefit medical diagnostic or procedural equipment with better imaging capabilities. If optical biopsy can be demonstrated as a diagnostic tool, this can lead to less unnecessary tissue removal for physical biopsies. I hope that this analysis is useful for new designs of optical instruments that incorporate active focusing mirrors, leveraging their precision, speed, and small size to build more functional and useful instruments for biomedicine and industrial imaging applications.

157

APPENDICES

158

APPENDIX A

VERIFICATION OF -FUNCTION WITH ZEMAX RAY TRACING RESULTS FOR

STOP AT MIRROR

159

Zemax ray tracing has a couple of main differences from the characteristic equation results ( -function) presented in this dissertation. The -function calculates the optical path difference between the chief ray and the ray of interest from point to point.

Zemax determines the optical path of the ray of interest until it reaches the image plane or

from point to plane. Figure 87 shows a spot diagram when the mirror is used at 20

o incidence angle with the object at infinity with no field. With only 5

 m deflection, the rays extend out to 5

 m from the chief ray at (0, 0). By simple geometry this leads to a difference in OPD of . Most likely, this error increases with greater deflections and greater angles of incidence.

(0, 0)

Figure 87: Zemax spot diagram of a 20 o

incidence angle 4 mm x 4.26 mm elliptical mirror with 5

 m deflection and an object at infinity with no field.

160

Additionally, the Hamiltonian aberration analysis assumes all rays of interest reflect off the MEMS mirror surface and fill the exit pupil. When the MEMS mirror is used in a system with strongly diverging rays, however, the rays may vignette and underfill the exit pupil. For example, with

2

= 45 o

,

r

1

= 10 mm, and

r

2

= -13.944 mm, vignetting occurs at the MEMS mirror surface on the side of the mirror furthest away

from the object and the rays never reach the exit pupil (Figure 88). The Hamiltonian

analysis assumes all rays reach the exit pupil, which is equivalent to allowing the MEMS aperture to increase to ensure all rays reflect off its surface. In Zemax, the RMS calculation provides 0.8867

 m and 1.17

 m for the under-filled and filled exit pupil, respectively. My analysis calculates 1.16

 m for the RMS for these conditions.

Object

Object

Exit pupil is under-filled.

Image

Image

Vignetting Un-constrained MEMS aperture

Figure 88: (left) For greatly divergent rays, the rays from the object miss the far edge of the MEMS mirror and never reach the exit pupil. (right) The Hamiltonian analysis assumes the exit pupil is filled or that the MEMS aperture may be extended to ensure all rays pass through the exit pupil and reach the image (

r

2

= -13.944 mm for the case shown).

161

The -function used to generate the aberration terms with stop at the mirror in

Chapter 4 is compared with Zemax OPD fans. It may be noted that Zemax normalizes the pupil coordinates and considers

y

-direction in the pupil plane to be the tangential plane regardless of optical system. The

y

-direction in the pupil plane for our system is the sagittal plane. Rays reflected from a mirror in Zemax result in negative

Z

-displacement, whereas it is defined as positive

Z

-displacement for the -function. The wavelength in

Zemax is set to 1

 m. A multitude of incidence angles and field conditions were verified, but only a few are presented here to aid the reader in having confidence in the results in this thesis. Zemax OPD fans for incidence angles of 0 o

, 5 o

, and 45 o

are compared with theoretical results from the -function.

Normal Incidence Mirror with No Field

Initially, I compare the -function for no-field imaging of a circular-boundary mirror ( ) with Zemax results (Table 22). Although not shown, the OPD fans in

Zemax and from the analytical analysis for a circular mirror with

r

1

= ∞ and no field both show agreement with values of zero across the pupil. Table 22 shows the constraints used to compare OPD fans for the case

r

1

= 200 mm and



= 20

m. Figure 89 and Figure 90

show agreement between Zemax and the analytical OPD fans. At 0.75 of the normalized pupil the OPD is 0.000949

 m.

162

Table 22: Constraints and values given to variables to compare Matlab code with Zemax.

Variable Relationship Imposed Value Assigned

Z lens r1

200 mm

r2



1

2 u2 v2 a

-

2

20

0

 o

0

0 m

2 mm

b

Figure 89: OPD fans in Zemax for circular mirror with normal incidence, no field,

= 20

 m, and

r

1

= 200 mm.

2.5

2

3 x 10

-3

V function

1.5

1

X: -0.75

Y: 0.0009492

0.5

163

3 x 10

-3

V function

2.5

2

1.5

1

X: -0.75

Y: 0.0009492

0.5

0

-1 -0.5

0 0.5

y in Pupil plane

1

0

-1 -0.5

0 0.5

x in Pupil plane

1

Figure 90: OPD fans from

= 20

 m, no field, and

r

1

-function

= 200 mm. for circular mirror with normal incidence,

5 o

Incidence Mirror with Small Field in

u

2

-direction

Table 23 shows the system that is compared: an

2

= 5 o

mirror with

u

2

= 0.5 mm and

= 2

 m. Values on the OPD fans in

x

- and

y

-directions show good agreement

(Figure 92 and Figure 93). At 0.75 on the normalized pupil in Zemax, the graphs provide values with magnitudes of 0.00195

 m, 0.000483

 m, and 0.000097

 m. Analytical results provide values with magnitudes of 0.001952

 m, 0.0004827

 m, and

0.000097

 m.

164

Table 23: Constraints and values given to variables to compare Matlab code with Zemax.

Variable Relationship Imposed Value Assigned

ZL r1

inf

r2



1

2 u2 v2 a

-

2

2

-5 o m

0.5 mm

0 mm

2 mm

b

Figure 91: OPD fans in Zemax for 5 o

incidence mirror,

v

2

= 0 mm, and

r

1

=

∞.



= 2

 m,

u

2

= 0.5 mm,

0 x 10

-4

-1

-2

X: -0.75

Y: -0.0001952

165

12 x 10

-4

10

8

V function

X: -0.75

Y: 0.0004827

6

4

2

0

-3

-2

X: 0.75

Y: -9.651e-05

V function

-1 -0.5

0 0.5

1

-4

-1 -0.5

0 0.5

1 y in Pupil plane x in Pupil plane

Figure 92: OPD fans from -function for 5 o

incidence mirror,

= 2

 m,

u v

2

= 0 mm, and

r

1

=

∞.

2

= 0.5 mm,

45 o

Incidence Mirror

Table 24 shows a system that is compared: an

2

= 45 o

mirror with no field and

= 20

m. Figure 93 provides details of the layout in Zemax. Values on the OPD fans in

x

- and

y

-directions show good agreement (Figure 94 and Figure 95). At 0.75 on the

normalized pupil in Zemax, the graphs provide values with magnitudes of 0.000447

 m,

0. 170031

 m, and 0.167346

 m. Analytical results provide values with magnitudes of

0.0004475

 m, 0.1756

 m, and 0.1622

 m.

166

Table 24: Constraints and values given to variables to compare Matlab code with Zemax.

Variable Relationship Imposed Value Assigned

ZL r1

inf

r2



1

2 u2 v2 a

-

2

20

 m

-44.9999

0

0

2 mm o

b

Figure 93: Lens Data Editor and 3D layout in Zemax for 45

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞.

o

incidence mirror,

167

Figure 94: OPD fans in Zemax for 45 o

incidence mirror,

v

2

= 0 mm, and

r

1

=

∞.



= 20

 m,

u

2

= 0 mm,

1.5

x 10

-3

V function

0.4

0.3

0.2

0.1

X: -0.75

Y: 0.1622

V function

1

0.5

X: -0.75

Y: 0.0004475

0

-0.1

-0.2

-0.3

-0.4

X: 0.75

Y: -0.1756

0

-1 -0.5

0 0.5

1

-0.5

-1 -0.5

0 0.5

1 y in Pupil plane x in Pupil plane

Figure 95: OPD fans from -function for 45 o

incidence mirror,



= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞.

168

Table 25 shows a system that is compared: an

2

= 45 o

mirror with

u

2

= 5 mm,

v

2

= 0 mm, and



= 20

m. Figure 96 provides details of the layout in Zemax. Values on

the OPD fans in

x

- and

y

-directions show good agreement (Figure 97 and Figure 98). At

0.75 on the normalized pupil in Zemax, the graphs provide values with magnitudes of

1.122332

 m, 1.113697

 m, and 0.825761

 m. Analytical results provide values with magnitudes of 1.121

 m, 1.081

 m, and 0.8486

 m.

Table 25: Constraints and values given to variables to compare Matlab code with Zemax.

Variable Relationship Imposed Value Assigned

ZL r1

inf

r2



1

2 u2 v2 a

-

2

20

 m

-44.9999

5 mm

0

2 mm o

b

169

Figure 96: Lens Data Editor and 3D Layout in Zemax for 45

= 20

 m,

u

2

= 5 mm,

v

2

= 0 mm, and

r

1

= ∞.

o

incidence mirror,

Figure 97: OPD fans from Zemax for 45 o

incidence mirror,

v

2

= 0 mm, and

r

1

= ∞.



= 20

 m,

u

2

= 5 mm,

170

-0.6

-0.8

-1

-1.2

-1.4

0

-0.2

-0.4

X: -0.75

Y: -1.121

1.4

1.2

1

0.8

0.6

2

1.8

1.6

X: -0.75

Y: 1.081

V function

X: 0.75

Y: 0.8486

-1.6

0.4

-1.8

0.2

V function

-2

-1 -0.5

0 0.5

y in Pupil plane

1

0

-1 -0.5

0 0.5

x in Pupil plane

1

Figure 98: OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 5 mm,

v

2

= 0 mm, and

r

1

= ∞.

Table 26 shows a system that is compared: an

2

= 45 o

mirror with

u

2

= 0 mm,

v

2

= 5 mm, and

= 20

m. Figure 99 provides details of the layout in Zemax. Values on

the OPD fans in

x

- and

y

-directions show good agreement (Figure 100 and Figure 101).

At 0.75 on the normalized pupil in Zemax, the graphs provide values with magnitudes of

0.103078

 m, 0.055711

 m, 0.089472

 m, and 0.245362

 m. Analytical results provide values with magnitudes of 0.09021

 m, 0.06676

 m, 0.08493

 m, and 0.2571

 m.

171

Table 26: Constraints and values given to variables to compare Matlab code with Zemax.

Variable Relationship Imposed Value Assigned

ZL r1

inf

r2



1

2 u2 v2 a

-

2

20

 m

-44.9999

0

5 mm

2 mm o

b

Figure 99: Lens Data Editor and 3D Layout in Zemax

= 20

 m,

u

2

= 0 mm,

v

2

= 5 mm, and

r

1

= ∞.

for 45 o

incidence mirror,

172

Figure 100: OPD fans from Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 5 mm, and

r

1

= ∞.

0

-0.02

-0.04

X: 0.75

Y: -0.06676

0.3

0.2

0.1

0

X: -0.75

Y: 0.08493

V function

-0.06

-0.08

-0.1

-0.12

-0.14

X: -0.75

Y: -0.09021

-0.1

-0.2

-0.3

-0.4

X: 0.75

Y: -0.2571

-0.16

-0.5

V function

-0.18

-1 -0.5

0 0.5

y in Pupil plane

1

-0.6

-1 -0.5

0 0.5

x in Pupil plane

1

Figure 101: OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 5 mm, and

r

1

= ∞.

173

Table 27 shows a system that is compared: an

2

= 45 o

mirror with no field,

= 20

 m, and

r

1

= 200 mm. All of the previous graphs for an

2

= 45 o

mirror were for

r

1

= ∞. Values on the OPD fans in

x

- and

y

-directions show good agreement (Figure 102 and Figure 103). At 0.75 on the normalized pupil in Zemax, the graphs provide values

with magnitudes of 0.000594

 m, 0.050923

 m, and 0.047945

 m. Analytical results provide values with magnitudes of 0.0008565

 m, 0.00503

 m, and 0.04856

 m.

Table 27: Constraints and values given to variables to compare Matlab code with Zemax.

Variable

Z lens r1

Relationship Imposed Value Assigned

200 mm

r2



1

2 u2 v2 a

-

2

20

-45

0

0 o m

2 mm

b

174

Figure 102: OPD fans from Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= 200 mm.

3 x 10

-3

V function

0.15

V function

0.1

2.5

X: -0.75

Y: 0.04856

0.05

2

0

1.5

X: 0.75

Y: -0.0503

-0.05

1

X: -0.75

Y: 0.0008565

-0.1

0.5

-0.15

0

-1 -0.5

0 0.5

1

-0.2

-1 -0.5

0 0.5

1 y in Pupil plane x in Pupil plane

Figure 103: OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= 200 mm.

175

Table 28 shows a system that is compared: an

2

= 45 o

mirror with

u

2

=

v

2

= 5 mm,

= 20

 m, and

r

1

= 200 mm. Values on the OPD fans in

x

- and

y

-

directions show good agreement (Figure 104 and Figure 105). At 0.75 on the normalized

pupil in Zemax, the graphs provide values with magnitudes of 0.767519

 m,

0.748501

 m, 0.679464

 m, and 0.581489

 m. Analytical results provide values with magnitudes of 0.76

 m, 0.7558

 m, 0.6589

 m, and 0.5987

 m.

Table 28: Constraints and values given to variables to compare Matlab code with Zemax.

Variable Relationship Imposed Value Assigned

Z lens r1

200 mm

r2



1

2 u2 v2 a

-

2

20

-45

5

5 o m

2 mm

b

176

Figure 104: OPD fans from Zemax for 45 o

incidence mirror,

= 20

 m,

u

2

= 5 mm,

v

2

= 5 mm, and

r

1

= 200 mm.

0

-0.2

-0.4

-0.6

-0.8

-1

X: -0.75

Y: -0.76

X: 0.75

Y: -0.7558

1.4

1.2

1

0.8

0.6

0.4

V function

X: -0.75

Y: 0.6589

X: 0.75

Y: 0.5987

-1.2

0.2

V function

-1.4

-1 -0.5

0 0.5

y in Pupil plane

1

0

-1 -0.5

0 0.5

x in Pupil plane

1

Figure 105: OPD fans from -function for 45 o

incidence mirror,

= 20

 m,

u

2

= 5 mm,

v

2

= 5 mm, and

r

1

= 200 mm.

177

APPENDIX B

VERIFICATION THAT 2 nd

and 3 rd

ORDER

TERMS ACCURATELY REPRESENT -FUNCTION

178

The difference between the entire -function and summing 2 nd

- and 3 rd

- order terms of the -function generally remains below 1% error for the case of stop at the mirror and paraboloidal shape of the mirror. One may find tables with conditions examined and plots of only the 2 nd

- order terms, the 2 nd

- and 3 rd

- order combined terms,

and the entire -function below for a few specific cases. Table 29 provides system details

for a parabolic mirror with 5 o

incidence angle,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. Figure 106 illustrates the contribution of different orders of terms of the

function. It also shows strong agreement between the entire -function and 2 nd

- and 3 rd

- order terms only.

Table 29: Constraints and values given to variables.

Variable Relationship Imposed

Z lens r1 r2



1

2 u2 v2 a

-

2

Value Assigned

inf

20

-5

0

0 o m

2 mm

b

179

Figure 106: (left) Comparison of total calculated -function, parabasal, 2 nd

-order aberration contribution, and 3

5 o

incidence angle, rd

- order aberration contribution for a parabolic mirror with

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. The optical path difference is shown for

x

in the pupil plane (tangential plane). (right) The optical path difference and Taylor series expansion of the -function are shown for

y

in the pupil plane (sagittal plane).

Table 30 provides system details for a parabolic mirror with 45

o

incidence angle,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. Figure 107 illustrates the contribution of

different orders of terms of the -function. One might note that 2 nd

- order terms primarily contribute to its aberration, whereas for 5 o

incidence angle the contribution of the 2 nd

- order terms were minimal. It also shows strong agreement between the entire -function and 2 nd

- and 3 rd

- order terms only.

180

Table 30: Constraints and values given to variables.

Variable Relationship Imposed

Z lens r1 r2



1

2 u2 v2 a

-

2

Value Assigned

inf

20

 m

-44.9999

0

0

2 mm o

b

Figure 107: (left) Comparison of total calculated rd aberration contribution, and 3

45 o

-function, parabasal, 2 nd

- order

- order aberration contribution for a parabolic mirror with

incidence angle,

= 20

 m,

u

2

= 0 mm,

v

2

= 0 mm, and

r

1

= ∞. The optical path difference is shown for

x

in the pupil plane (tangential plane). (right) The optical path difference and Taylor series expansion of the -function are shown for

y

in the pupil plane (sagittal plane).

181

APPENDIX C

ZERNIKE DEFINITIONS

182

Matlab generated Zernike fits to surface profiles at normal incidence in reflectance mode. Fricker’s zernfun2.m and zernpol.m are used [122]. The wavefront sensor generated Zernike fits for measured experimental results. Zemax Zernike coefficients are used for general comparison with the experimental results. Table 31 lists rectangular Zernike terms and how they are defined for all three cases mentioned above.

Zemax and Matlab map onto a circular exit pupil and use equivalent polar coordinates for their circularly symmetric fit. Matlab Zernike coefficients are not normalized, whereas

Zemax and the WFS have normalized coefficients.

Table 31: List of Zernike term definitions.

183

Matlab

Mode*'

1

Zemax

Mode*

1

Wavefront

Sensor Mode (j)

1

Order

(n)

0

Frequency

(m)

0

Normalization

Factor

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

3

2

5

4

6

9

7

8

10

15

13

11

12

14

21

19

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

1

1

2

2

2

3

3

3

3

4

4

4

4

4

5

5

-1

1

-2

0

2

-3

-1

1

3

-4

-2

0

2

4

-5

-3

√8

√10

√10

√5

√10

√10

√12

√12

√6

√8

√8

√8

2

2

√6

√3

(6X

(15X

4

(4X

(5X

(2X

(X

(4X

(8X

4

+12X

4

(X

2

4

Y+10X

2

3

3

3

2XY

2

(X

(3X

(3X

(3X

Y

2

Y

X

+2Y

2

2

-Y

Y-Y

3

2

Y+3Y

3

+3XY

)

3

-2Y)

2

-2X)

-3XY

Y-4XY

Y+4XY

Y

2

+6Y

4

-4Y

4

-6X

-3X

2

Y

4

Y-10X

2

2

3

-5Y

2

)

2

2

-1)

Y

3

)

+Y

3

3

4

)

)

-6X

2

+3Y

)

+Y

-6Y

2

5

-12X

2

5

2

)

)

+1)

Y+4Y

3

)

18

19

20

21

22

23

24

17

16

18

20

27

25

23

18

19

20

21

22

23

24

5

5

5

5

6

6

6

-1

1

3

5

-6

-4

-2

√12

√12

√12

√12

√14

√14

√14

(10X

(10X

(30X

5

4

5

Y+20X

+20X

(5X

5

+10X

(24X

3

Y

(X

(6X

Y+60X

3

2

3

5

2

Y

3

+10Y

+10XY

Y

2

-15XY

4

-10X

3

5

Y-24XY

Y

3

5

Y-20X

Y

5

4

2

-12X

-12X

+5XY

5

-20X

+30XY

5

3

Y

3

2

3

-4X

3

Y-12Y

-12XY

4

+12XY

2

+6XY

3

)

Y+20XY

-40X

3

5

)

3

3

2

Y-40XY

3

+3Y)

+3X)

)

)

+12XY)

25 22 25

6 0

√7

(20X

6

+60X

4

Y

2

+60X

2

Y

4

+20Y

6

-30X

4

-60X

2

Y

2

-30Y

4

+12X

2

+12Y

26

24

26 6 2

√14

(15X

*Note: Zemax and Matlab use equivalent polar coordinates and assume a circularly symmetric fit

6

+15X

4

Y

2

-15X

2

Y

4

-15Y

6

-20X

4

+20Y

4

+6X

2

-6Y

2

)

2

-1)

'Note: Matlab Zernike coefficients are not normalized.

184

APPENDIX D

ELECTROSTATIC PRESSURE VERSUS DISPLACEMENT

185

I develop the relationship between the electrostatic pressure due to a voltage

across the mirror and the mirror’s maximum center displacement. Figure 108 shows the

geometry of an elliptical-boundary SU-8 2002 membrane with minor radius

a

, thickness

h

SU-8

, and permittivity

1

. In-plane tension

T

[N/m] suspends the membrane above a fluid medium with permittivity

2

. A layer of oxide with thickness

h oxide

exists below the fluid medium with permittivity

3

. A conductive film at the top supports a voltage

V

. We assume the bottom silicon electrode maintains a ground voltage,

V

= 0. The charge densities in the top conductive film, at the bottom of the membrane, and between the oxide and the fluid medium are

q

1

(X, Y)

[C/m

2

],

q

2

(X, Y)

[C/m

2

], and

q

3

(X, Y)

[C/m

2

], respectively. Deflection of the membrane into the initial air gap

s o

is described by shape

s(X, Y).

Assume the membrane, fluid medium, and oxide are non-conducting, the gap is small compared to the radial extent of the membrane, and the electric fields

E

1

, E

2

, and

E

3

are everywhere directed.

h

1 s o h

3 q

1

(X,Y)

E

1 q

2

(X,Y)

E

2

E

3 q

3

(X,Y)

X a

Z s(X,Y)

V

=

V

1

2

3

Y

V

= 0

Figure 108: Electrostatic pressure schematic.

Using Gauss’s law, where the total outward flux through the surfaces of a volume must equal the total charge enclosed, charge densities at the boundaries may be

186 determined. The charge densities and applied voltage can be related to the electric fields as follows, and .

Assuming that

q

2

= 0 and

q

3

= 0 or that the membrane and oxide layer hold no charge,

.

This provides

. (38)

The equation for stored energy in a capacitor is , where

C

[F] is the capacitance and

Q

[C] is the charge. The equation for capacitance of a parallel-plate capacitor is , where

A

is the area of the plates and

d

is the distance between plates. Substituting in the series capacitance of the three dielectrics, the stored energy for the membrane with constant charge is,

(39)

The electrostatic force is defined as [N]. We are only concerned with the direction and can, therefore, analyze the problem as a scalar. Taking the derivative of

W e

with respect to

s(X, Y)

and dividing by

A

, the electrostatic pressure is

[N/m

2

]. In terms of the applied voltage this is

.

(40)

The equation of motion for a membrane is

187

(41) where

[kg/m

2

] is the mass density of the membrane [148]. Tension

T

is related to the intrinsic stress

of the membrane as follows,

.

(42)

The intrinsic stress of the membrane varies with process parameters. Under equilibrium conditions assuming , the equation of motion for a membrane reduces to

.

Using the Laplacian operator, the membrane motion may be described by

(43)

. (44)

Equation (14) with

C a2

=

may be substituted in for

s(X,Y)

in Equation 44, resulting in

.(45)

Assuming small deflections

s(X, Y)

<<

s o

, this may be simplified to

(46)

The intrinsic stress values in Table 1 (Chapter 1) are calculated with values of 3.9 [121]

and 4.1 [114] for the relative permittivity of thermal oxide and SU-8 2002, respectively.

188

REFERENCES CITED

189

[1] Lukes, S. J. and Dickensheets, D. L. "SU-8 2002 surface micromachined deformable membrane mirrors,"

Microelectromechanical Systems, Journal of

22(1), 94-

106 (2013).

[2] Lutzenburger, B. J., Moghimi, M. J., Lukes, S. J., Kaylor, B., and Dickensheets,

D. L. "MEMS deformable mirrors for focus control in vital microscopy,"

SPIE MOEMS and Miniaturized Systems IX

, 759406-8 (2010).

[3] Booth, M. J. "Adaptive optics in microscopy,"

Philosophical Transactions of the

Royal Society A: Mathematical, Physical and Engineering Sciences

365(1861), 2829-

2843 (2007).

[4] Schwertner, M., Booth, M. J., and Wilson, T. "Adaptive optics for microscopy, optical data storage, and micromachining." In

SPIE Optics+ Photonics

(pp. 63060A-

63060A). International Society for Optics and Photonics. (2006).

[5] Kang, D., Yoo, H., Jillella, P., Bouma, B. E., and Tearney, G. J. "Comprehensive volumetric confocal microscopy with adaptive focusing,"

Biomed. Opt. Express

2(6),

1412-1422 (2011).

[6] Chu, K., Smith, Z. J., Dwyre, D., Matthews, D., Lane, S., and Wachsmann-Hogiu,

S. "Microscopy and spectroscopy on a cell phone,"

Bio-Optics: Design and Application

,

BWA3 (2011).

[7] Sherman, L., Ye, J. Y., Albert, O., and Norris, T. B. "Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,"

Journal of Microscopy

206(1), 65-71 (2002).

[8] Botcherby, E. J., Juškaitis, R., Booth, M. J., and Wilson, T. "An optical technique for remote focusing in microscopy,"

Optics Communications

281(4), 880-887

(2008).

[9] Tsai, P. S., Migliori, B., Campbell, K., Kim, T. N., Kam, Z., Groisman, A., and

Kleinfeld, D. "Spherical aberration correction in nonlinear microscopy and optical ablation using a transparent deformable membrane,"

Applied Physics Letters

91(19),

191102 (2007).

[10] Fernandez, E. and Artal, P. "Membrane deformable mirror for adaptive optics:

Performance limits in visual optics,"

Opt. Express

11(9), 1056-1069 (2003).

[11] Dalimier, E. and Dainty, C. "Comparative analysis of deformable mirrors for ocular adaptive optics,"

Opt. Express

13(11), 4275-4285 (2005).

190

[12] Devaney, N., Dalimier, E., Farrell, T., Coburn, D., Mackey, R., Mackey, D.,

Laurent, F., Daly, E., and Dainty, C. "Correction of ocular and atmospheric wavefronts:

A comparison of the performance of various deformable mirrors,"

Appl. Opt.

47(35),

6550-6562 (2008).

[13] Fernández, E. J. and Artal, P. "Study on the effects of monochromatic aberrations in the accommodation response by using adaptive optics,"

J. Opt. Soc. Am. A

22(9),

1732-1738 (2005).

[14] Tuohy, S., Bradu, A., Podoleanu, A. G., and Chateau, N. "Correcting ocular aberrations with a high stroke deformable mirror." In

European Conference on

Biomedical Optics

, p. 66271L. International Society for Optics and Photonics (2007).

[15] Dickensheets, D. L., Moghimi, M.J., Lukes, S.J., Lutzenberger, B.J., Kaylor,

B.M., Schoessler, J., and Roos, P.A. "A compact F/5 camera lens with MEMS focus control," ICOME2011 Conference (2011).

[16] Hsieh, H.-T., Wei, H.-C., Lin, M.-H., Hsu, W.-Y., Cheng, Y.-C., and Su, G.-D. J.

"Thin autofocus camera module by a large-stroke micromachined deformable mirror,"

Opt. Express

18(11), 11097-11104 (2010).

[17] Wei, H. C., Chien, Y. H., Hsu, W. Y., Cheng, Y. C., and Su, G. D. J. "Controlling a MEMS deformable mirror in a miniature auto-focusing imaging system,"

Systems Technology, IEEE Transactions on

20(6), 1592-1596 (2012).

Control

[18] Kuiper, S., Hendriks, B. H. W., Suijver, J. F., Deladi, S., and Helwegen, I. "Zoom camera based on liquid lenses." In

Proc. SPIE

, vol. 6466, p. 64660F (2007).

[19] Kuiper, S. and Hendriks, B. H. W. "Variable-focus liquid lens for miniature cameras,"

Applied Physics Letters

85(7), 1128-1130 (Aug 2004).

[20] Wang, J.-L., Chen, T.-Y., Chien, Y.-H., and Su, G.-D. J. "Miniature optical autofocus camera by micromachined fluoropolymer deformable mirror,"

Opt. Express

17(8), 6268-6274 (2009).

[21] Gutierrez, R. C., Tang, T. K., Calvet, R., and Fossum, E. R. "MEMS digital camera,"

Digital Photography III

, 6502, 65020K-8 (2007).

[22] Kinoshita, H., Hoshino, K., Matsumoto, K., and Shimoyama, I. "A thin camera with a zoom function using reflective optics,"

Sensors and Actuators a-Physical

128(1),

191-196 (2006).

[23] Zhao, X., Xie, Y., and Zhao, W. "Broadband and wide field of view foveated imaging system in space,"

Optical Engineering

47(10), 103202 (2008).

191

[24] Park, J. H., Garipov, G. K., Jeon, J. A., Khrenov, B. A., Kim, J. E., Kim, M., Kim,

Y. K., Lee, C. H., Lee, J., Na, G. W., Nam, S., Park, I. H., and Park, Y. S. "Obscura telescope with a MEMS micromirror array for space observation of transient luminous phenomena or fast-moving objects,"

Optics Express

16(25), 20249-20257 (2008).

[25] Po-Yu, L., et al. "Design and fabrication of a large-stroke MEMS deformable mirror for wavefront control,"

Journal of Optics

13(5), 055404.

[26] Peter, Y. A., Gonte, F., Herzig, H.-P., and Dandliker, R. "Micro-optical fiber switch for a large number of interconnects using a deformable mirror,"

Photonics

Technology Letters, IEEE

14(3), 301-303 (2002).

[27] Xiaohua, M. and Kuo, G. S. "Optical switching technology comparison: Optical

MEMS vs. other technologies,"

Communications Magazine, IEEE

41(11), S16-S23

(2003).

[28] Chu, P. B., Brener, I., Pu, C., Lee, S. S., Dadap, J. I., Park, S., Bergman, K.,

Bonadeo, N. H., Chau, T., Chou, M., Doran, R. A., Gibson, R., Harel, R., Johnson, J. J.,

Lee, C. D., Peale, D. R., Tang, B., Tong, D. T. K., Tsai, M. J., Wu, Q., Zhong, W.,

Goldstein, E. L., Lin, L. Y., and Walker, J. A. "Design and nonlinear servo control of

MEMS mirrors and their performance in a large port-count optical switch,"

J. of

Microelectromechanical Systems

14(2), 261-273 (2005).

[29] Bonora, S., Brida, D., Villoresi, P., and Cerullo, G. "Ultrabroadband pulse shaping with a push-pull deformable mirror,"

Opt. Express

18(22), 23147-23152 (2010).

[30] Bonora, S., Bortolozzo, U., Naletto, G., and Residori, S. "Innovative membrane deformable mirrors,"

Topics in Adaptive Optics

(2012).

[31] Rondi, A., Extermann, J., Bonacina, L., Weber, S. M., and Wolf, J. P.

"Characterization of a MEMS-based pulse-shaping device in the deep ultraviolet,"

Applied Physics B

96(4), 757-761 (2009).

[32] Cheng, X., Ma, J., and Hao, Q. "Dynamic focus control of the Blu-ray optical pickup unit,"

Optical Design and Engineering III

, 7100, 71000V-8 (2008).

[33] Fanget, S., Labeye, P. R., Divoux, C., and Hugon, X. "Integrated deformable mirror on silicon for optical data storage,"

MOEMS Display and Imaging Systems III

,

5721, 159-169 (2005).

[34] Aoki, S., Yamada, M., and Yamagami, T. "A novel deformable mirror for spherical aberration compensation,"

Jpn. J. Appl. Phys.

48, 03A003 (2009).

192

[35] Beck, R. J., Parry, J. P., MacPherson, W. N., Waddie, A., Weston, N. J.,

Shephard, J. D., and Hand, D. P. "Application of cooled spatial light modulator for high power nanosecond laser micromachining,"

Opt. Express

18(16), 17059-17065 (2010).

[36] Mansell, J. D., Henderson, B. G., and Robertson, G. "Evaluation of polymer membrane deformable mirrors for high peak power laser machining applications,"

Proc.

SPIE

, 7816, 78160D (2010).

[37] Wang, F. "Utility transforms of optical fields employing deformable mirror," Opt.

Lett. 36(22), 4383-4385 (2011).

[38] Huang, Y., Wan, J., Cheng, M.-C., Zhang, Z., Jhiang, S. M., and Menq, C.-H.

"Three-axis rapid steering of optically propelled micro/nanoparticles,"

Scientific Instruments

80(6), 063107-8 (2009).

Review of

[39] Lukes, S. J.

Surface Micro-machined SU-8 2002 Deformable Membrane Mirrors

, master’s thesis in electrical engineering, Bozeman: Montana State University (2011).

[40] Lukes, S. J. and Dickensheets, D. L. "SU-8 focus control mirrors released by

XeF2 dry etch,"

SPIE MOEMS and Miniaturized Systems X

, 7930, 793006-6 (2011).

[41] Himmer, P. A., Dickensheets, D. L., and Friholm, R. A. "Micromachined silicon nitride deformable mirrors for focus control,"

Opt. Letters

26(16), 1280-1282 (2001).

[42] Moghimi, M. J., Lutzenberger, B. J., Kaylor, B. M., and Dickensheets, D. L.

"MOEMS deformable mirrors for focus control in vital microscopy,"

Journal of

Micro/Nanolithography, MEMS and MOEMS

10(2), 023005-8.

[43] Korde, U. A. "Large-displacement closed-loop control of variable area electrostatic actuation for membrane reflectors,"

Journal of Intelligent Material Systems and Structures

20(6), 697-721 (2009).

[44] Lin, M.-J. and Wu, K.-W. "Design and fabrication of an electrostatically actuated microdeformable focusing mirror,"

MOEMS

10(1), 011504-7 (Mar 2011).

Journal of Micro/Nanolithography, MEMS and

[45] Mansell, J. D., Henderson, B. G., and Robertson, G. "Evaluation of polymer membrane deformable mirrors for high peak power laser machining applications."

In

SPIE Optical Engineering+ Applications

(pp. 78160D-78160D). International Society for Optics and Photonics (2010).

[46] Wang, J. L., Chen, T. Y., Liu, C., Chiu, C. W. E., and Su, G. D. J. "Polymer deformable mirror for optical auto focusing,"

ETRI Journal

29(6), 817-819 (Dec 2007).

193

[47] Wei, H.-C. and Su, G.-D. J. "A low voltage deformable mirror using ionicpolymer metal composite,"

Proc. SPIE

7788, 7788, 77880C (2010).

[48] Bonora, S., Coburn, D., Bortolozzo, U., Dainty, C., and Residori, S. "High resolution wavefront correction with photocontrolled deformable mirror,"

Opt. Express

20(5), 5178-5188 (2012).

[49] Berge, B. "Liquid lens technology: Principle of electrowetting based lenses and applications to imaging,"

18th IEEE International Conference on MEMS

, 227-230

(2005).

[50] Werber, A. and Zappe, H. "Tunable pneumatic microoptics,"

JMEMS

17(5),

1218-1227 (2008).

[51] Yang, H., Yang, C.-Y., and Yeh, M.-S. "Miniaturized variable-focus lens fabrication using liquid filling technique,"

Microsystem Technologies

14(7), 1067-1072

(May 2008).

[52] Lee, S. W. and Lee, S. S. "Focal tunable liquid lens integrated with an electromagnetic actuator,"

Applied Physics Letters

90(12), 121129 (Mar 2007).

[53] Hendriks, B. H. W., Kuiper, S., Van As, M. A. J., Renders, C. A., and Tukker, T.

W. "Electrowetting-based variable-focus lens for miniature systems,"

Optical Review

12(3), 255-259 (2005).

[54] Carpi, F., Frediani, G., Turco, S., and De Rossi, D. "Bioinspired tunable lens with muscle-like electroactive elastomers,"

Advanced Functional Materials

21(21), 4152-4158

(2011).

[55] Dubois, P., Rosset, S., Koster, S., Stauffer, J., Mikhaïlov, S., Dadras, M., Rooij,

N.-F., d., and Shea, H. "Microactuators based on ion implanted dielectric electroactive polymer (EAP) membranes,"

(2006).

Sensors and Actuators A: Physical

130-131, 147-154

[56] Dubois, P., Rosset, S., Niklaus, M., Dadras, M., and Shea, H. "Voltage control of the resonance frequency of dielectric electroactive polymer (DEAP) membranes,"

J. of

Microelectromechanical Systems

17(5), 1072-1081 (2008).

[57] Rosset, S., Niklaus, M., Dubois, P., and Shea, H. R. "Mechanical characterization of a dielectric elastomer microactuator with ion-implanted electrodes,"

Actuators A: Physical

144(1), 185-193 (2008).

Sensors and

[58] e-Vision. "Lens Systems for Vision Correction-electronic Eyewear and IOLS," http://evisionoptics.com/spectacle.html Accessed (June 23, 2013).

194

[59] Martinez, T., Wick, D. V., Payne, D. M., Baker, J. T., and Restaino, S. R. "Nonmechanical zoom system,"

SPIE Sensors, Systems, and Next-Generation Satellites VII

,

5234, 375-378 (2004).

[60] Wick, D. V., Bagwell, B. E., Sweatt, W. C., Peterson, G. L., Martinez, T.,

Restaino, S. R., Andrews, J. R., Wilcox, C. C., Payne, D. M., and Romeo, R. "Active optical zoom for space-based imaging,"

SPIE Unconventional Imaging II

, 63070A

(2006).

[61] Stockbridge, C., Lu, Y., Moore, J., Hoffman, S., Paxman, R., Toussaint, K., and

Bifano, T. "Focusing through dynamic scattering media,"

Opt. Express

20(14), 15086-

15092 (2012).

[62] Johnson, R. C. "MEMS Market to Top $22 billion by 2018," EE Times, http://www.eetimes.com/document.asp?doc_id=1320035 Accessed (11/14/2013).

[63]

Tablets

Wood, L. "Research and Markets: MEMS & Sensors for Mobile Phones and

2013-2019," Business Wire, Last Updated: June 26, 2014, http://www.businesswire.com/news/home/20140626005504/en/Research-Markets-

MEMS-Sensors-Mobile-Phones-Tablets#.VBZ1KJRxXTo Accessed (September 14).

[64] Graham-Rowe, D. "Liquid lenses make a splash," Nat Photon sample (sample), 2-

4 (2006).

[65] "Deformable Mirrors," Adaptica, deformable-mirrors Accessed (June 24, 2013).

[66] "Standard Deformable Mirrors," http://www.adaptica.com/site/en/pages/

Boston Micromachines Corporation, http://www.bostonmicromachines.com/production_products.htm Accessed (June 24,

2013).

[67] Imagine Optic, “Adaptive Optics Components & Software,” http://www.imagineoptic.com/iop_products_adaptive-optics_main_en.php Accessed (2011).

[68] ALPAO, “Deformable Mirrors,” http://www.alpao.fr/products_DM52.html

Accessed (2011).

[69] Flexible Optical B.V., "AO Systems with MMDM," http://www.okotech.com/aosystems-with-membrane-mirrors Accessed (June 24, 2013).

[70] Lukes, S. J. and Dickensheets, D. L. "MEMS focus control and spherical aberration correction for multilayer optical discs." In

SPIE MOEMS-MEMS

(pp.

82520L-82520L). International Society for Optics and Photonics (2012).

195

[71] Oliver, K. W., Lukes, S. J., Moghimi, M. J., and Dickensheets, D. L. "Stress engineering for free-standing SU-8 2002 thin film devices." In

SPIE MOEMS-MEMS

(pp.

82480H-82480H). International Society for Optics and Photonics (2012).

[72] Lukes, S. J., Himmer, P. A., Moog, E. J., Shaw, S. R., and Dickensheets, D. L.

"Feedback-stabilized deformable membrane mirrors for focus control,"

Micro/Nanolithography, MEMS, and MOEMS

8(4), 043040 (7 pp.) (2009).

J. of

[73] Lukes, S. J., Lutzenberger, B. J., Dunbar, E., Shaw, S. R., and Dickensheets, D. L.

"Variable-focus SU-8 membrane mirror with enhanced stroke using feedback control"

IEEE/LEOS Intl. Conf. on Optical MEMs and Nanophotonics

(2009).

[74] Wang, J.-L., Chen, T.-Y., Chien, Y.-H., and Su, G.-D. J. "Miniature optical autofocus camera by micromachined fluoropolymer deformable mirror,"

Optics Express

17(8), 6268-6274 (2009).

[75] Tsai, S.-A., Wei, H.-C., and Su, G.-D. J. "Polydimethylsiloxane coating on an ionic polymer metallic composite for a tunable focusing mirror,"

Appl. Opt.

51(35),

8315-8323 (2012).

[76] Hokari, R. and Hane, K. "A varifocal convex micromirror driven by a bending moment,"

Selected Topics in Quantum Electronics, IEEE Journal of

15(5), 1310-1316

(2009).

[77] Lukes, S. J. and Dickensheets, D. L. "MEMS mirror for flexible z-axis control in a commercial confocal microscope,"

Optical MEMS and Nanophotonics (OMN), 2012

International Conference on

, 148-149 (2012).

[78] Zhu, L., Sun, P.-C., Bartsch, D.-U., Freeman, W. R., and Fainman, Y. "Wavefront generation of Zernike polynomial modes with a micromachined membrane deformable mirror,"

Appl. Opt

. 38(28), 6019-6026 (1999).

[79] Booth, M., Wilson, T., Sun, H.-B., Ota, T., and Kawata, S. "Methods for the characterization of deformable membrane mirrors,"

Appl. Opt

. 44(24), 5131-5139 (2005).

[80] Booth, M. J., Schwertner, M., Wilson, T., Nakano, M., Kawata, Y., Nakabayashi,

M., and Miyata, S. "Predictive aberration correction for multilayer optical data storage,"

Applied Physics Letters

88(3), 031109-3 (2006).

[81] Dickensheets, D. L. "Requirements of MEMS membrane mirrors for focus adjustment and aberration correction in endoscopic confocal and optical coherence tomography imaging instruments,"

Journal of Microlithography, Microfabrication, and

Microsystems

7(2), 021008 (9 pp.) (Sep 2008).

196

[82] Simmonds, R. D., Wilson, T., and Booth, M. J. "Effects of aberrations and specimen structure in conventional, confocal and two-photon fluorescence microscopy,"

Journal of Microscopy

245(1), 63-71 (2012).

[83] Howard, J. M. and Stone, B. D. "Anamorphic imaging with three conic mirrors,"

International Optical Design Conference

, ITuA5 (2010).

[84] Delano, E. "First-order design and the y, y¯ diagram,"

Appl. Opt.

2(12), 1251-

1256 (1963).

[85] Lu, L.-J. and Deng, Z.-Y. "Geometric characteristics of aberrations of planesymmetric optical systems,"

Appl. Opt

. 48(36), 6946-6960 (2009).

[86] Howard, J. M. "Unobstructed spherical mirror systems," in

ProQuest

Dissertations and Theses

, United States -- New York: The University of Rochester

(2000).

[87] Howard, J. and Stone, B. D. "Nonanamorphic imaging with three conic mirrors,"

International Optical Design Conference

, IMA3 (2002).

[88] Howard, J. M. and Stone, B. D. "Imaging a point to a line with a single spherical mirror,"

Appl. Opt

. 37(10), 1826-1834 (1998).

[89] Stone, B. D. and Forbes, G. W. "Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,"

J. Opt. Soc. Am. A

11(12), 3292-3307

(1994).

[90] Choi, N., Kim, M., and Kim, J. "Approach to investigating the spherical aberration variation with the object position in an aplanatic system,"

Appl. Opt

. 48(2),

242-249 (2009).

[91]

[92]

Wynne, C. G. "Field correctors for parabolic mirrors,"

Physical Society. Section B

62(12), 772 (1949).

Proceedings of the

Mahajan, V. N. "Optical Imaging and Aberrations: Part 1. Ray Geometrical

Optics," Bellingham: SPIE (1998).

[93] Magdalena, M.-M., et al. "Wavefronts, light rays and caustic of a circular wave reflected by an arbitrary smooth curve,"

Journal of Optics

13(5), 055705.

[94] Lewis, D. A. "Optical characteristics of the whipple observatory TeV gamma-ray imaging telescope,"

Experimental Astronomy

1(4), 213-226 (1990).

197

[95] Ruze, J. "Lateral-feed displacement in a paraboloid,"

Antennas and Propagation,

IEEE Transactions on

13(5), 660-665 (1965).

[96] Saha, T. T. "Transverse ray aberrations for paraboloid-hyperboloid telescopes,"

Appl. Opt.

24(12), 1856-1863 (1985).

[97] Wynne, C. G. "Afocal Correctors for Paraboloidal Mirrors,"

1227-1231 (1967).

Appl. Opt

. 6(7),

[98] Arguijo, P. and Scholl, M. S. "Exact ray-trace beam for an off-axis paraboloid surface,"

Appl. Opt

. 42(16), 3284-3289 (2003).

[99] Magner, T. J., Zaniewski, J., Rice, S., and Fleetwood, C. "Fabrication and testing of off-axis parabolic mirrors,"

Optics & Laser Technology

19(2), 91-96 (1987).

[100] Bell, C. G., Ockendon, H., and Ockendon, J. R. "The caustics of two-Â and threedimensional parabolic reflectors,"

Journal of Optics

12(6), 065703.

[101] Brückner, C., Notni, G., and Tünnermann, A. "Optimal arrangement of 90° offaxis parabolic mirrors in THz setups,"

Electron Optics

121(1), 113-119.

Optik - International Journal for Light and

[102] Howard, J. E. "Imaging properties of off-axis parabolic mirrors,"

Applied Optics

18(15), 2714-2722 (1979).

[103] Murphy, J. A. "Distortion of a simple Gaussian beam on reflection from off-axis ellipsoidal mirrors,"

International Journal of Infrared and Millimeter Waves

8(9), 1165-

1187 (1987).

[104] Stone, B. D. and Forbes, G. W. "Illustration of second-order design methods:

Global merit function plots for a class of projection systems,"

J. Opt. Soc. Am. A

11(12),

3308-3321 (1994).

[105] Himmer, P. A. and Dickensheets, D. L. "Off-axis variable focus and aberration control mirrors,"

SPIE MOEMS and Miniaturized Systems III

, 4983, 296-303 (2003).

[106] Himmer, P. A. and Dickensheets, D. L. "Off-axis variable focus and aberration control mirrors and method," U.S. Patent 7,494,233 B2; United States: Montana State

University (Bozeman, MT), (2009).

[107] Yuhe, S., Dickensheets, D. L., and Himmer, P. "3-D MOEMS mirror for laser beam pointing and focus control,"

Selected Topics in Quantum Electronics, IEEE Journal of

10(3), 528-535 (2004).

198

[108] Buchdahl, H. A. "Hamiltonian optics: The point characteristic of a refracting plane,"

J. Opt. Soc. Am

. 60(8), 997-1000 (1970).

[109] Dickensheets, D. L. "Imaging performance of off-axis planar diffractive lenses,"

J. Opt. Soc. Am. A

13(9), 1849-1858 (1996).

[110] Mahajan, V. N.

Aberration Theory Made Simple, Tutorial Texts in Optical

Engineering

. Bellingham, Wash., USA: SPIE Optical Engineering Press (1991).

[111] Planchon, T. A., Mercere, P., Cheriaux, G., and Chambaret, J.-P. "Off-axis aberration compensation of focusing with spherical mirrors using deformable mirrors,"

Optics Communications

216(1), 25-31 (2003).

[112] Friese, C. and Zappe, H. "Deformable polymer adaptive optical mirrors,"

JMEMS

17(1), 11-19 (2008).

[113] Troesch, B. A. and Troesch, H. R. "Eigenfrequencies of an elliptic membrane,"

Mathematics of Computation

27(124), 755-765 (1973).

[114] MicroChem Corp, “Innovative Chemical Solutions for MEMS and

Microelectronics,” www.microchem.com Accessed (2009).

[115] Lukes, S. J. and Dickensheets, D. L. "Agile scanning using a MEMS focus control mirror in a commercial confocal microscope,"

SPIE BiOS

, 89490W-89490W-11 (2014).

[116] Yanhui, B., Yeow, J. T. W., Constantinou, P., Damaskinos, S., and Wilson, B. C.

"A 2-D micromachined SOI MEMS mirror with sidewall electrodes for biomedical imaging,"

Mechatronics, IEEE/ASME Transactions on

15(4), 501-510 (2010).

[117] Kronast, W., Mescheder, U., Müller, B., and Huster, R. "Development of a focusing micromirror device with an in-plane stress relief structure in silicon-on-insulator technology,"

J. of Micro/Nanolithography, MEMS, and MOEMS

13(1), 011112-011112

(2014).

[118] Keller, S., Blagoi, G., Lillemose, M., Haefliger, D., and Boisen, A. "Processing of thin SU-8 films,"

J. Micromech. Microeng

. 18(12), 125020 (10 pp.) (2008).

[119] Himmer, P. A. Silicon Nitride Deformable Mirrors for Focus and Spherical

Aberration Correction in Micro-optical Systems," PhD dissertation in electrical engineering, Bozeman, MT: Montana State University (2002).

[120] Shaw, M., Hall, S., Knox, S., Stevens, R., and Paterson, C. "Characterization of deformable mirrors for spherical aberration correction in optical sectioning microscopy,"

Opt. Express 18(7), 6900-6913 (2010).

199

[121] Gray, P. R., Hurst, P. J., Lewis, S. H., and Meyer, R. G.

Analysis and Design of

Analog Integrated Circuits

, 5th ed New York: John Wiley & Sons (2009).

[122] Fricker, P. "Zernike Polynomials," MATLAB Central File Exchange, Last

Updated: 11 Mar 2008, http://www.mathworks.com/matlabcentral/fileexchange/7687- zernike-polynomials Accessed (November 2010).

[123] Association, B.-r. D. "White Paper: Blu-ray Disc

TM

Format: General," Universal

City (2010).

[124] Ichimura, I., Saito, K., Yamasaki, T., and Osato, K. "Proposal for a multilayer read-only-memory optical disk structure,"

Appl. Opt

. 45(8), 1794-1803 (2006).

[125] Mitsumori, A., Higuchi, T., Yanagisawa, T., Ogasawara, M., Tanaka, S., and Iida,

T. "Multilayer 500 Gbyte optical disk,"

Japanese Journal of Applied Physics

48(3),

03A055 (2009).

[126] Kim, T. K., Ahn, Y. M., Kim, S. J., Heor, T. Y., Chung, C. S., and Park, I. S.

"Blu-ray disc pickup head for dual layer,"

Jpn. J. Appl. Phys

. 44(5B), 3397-3401 (2005).

[127] Katayama, R. and Komatsu, Y. "Blue/DVD/CD compatible optical head,"

Appl.

Opt.

47(22), 4045-4054 (2008).

[128] Komma, Y., Tanaka, Y., and Mizuno, S. "Compatible objective lens for blu-ray disc and digital versatile disc using diffractive optical element and phase-step element which corrects both chromatic and spherical aberrations,"

Japanese Journal of Applied

Physics

43(7B), 4768-4771 (2004).

[129] Tanaka, Y., Komma, Y., Shimizu, Y., Shimazaki, T., Murata, J., and Mizuno, S.

"Lens design of compatible objective lens for blu-ray disc and digital versatile disk with diffractive optical element and phase steps,"

43(7B), 4742-4745 (2004).

Japanese Journal of Applied Physics

[130] Morishita, I., Shindo, H., Takeya, N., Jeong, H., Yoon, Y., Chang, I., Kim, H.,

Lee, D., and Kyong, C. "Blu-ray disc/digital versatile disc recording and reproducing compatible use technology in the 2nd generation pick up for blu-ray disc,"

Japanese

Journal of Applied Physics

43(7B), 4746-4751 (2004).

[131] Hsi-Fu, S., Wei-Chih, L., and Jenq-Yang, C. "Design of single-path optical pickup head with three wavelengths using integrated optical unit,"

Magnetics, IEEE

Transactions on

45(5), 2202-2205 (2009).

200

[132] Katayama, R., Komatsu, Y., and Yamanaka, Y. "Dual-wavelength optical head with a wavelength-selective filter for 0.6- and 1.2-mm-thick-substrate optical disks,"

Appl. Opt

. 38(17), 3778-3786 (1999).

[133] Aoki, S., Yamada, M., and Yamagami, T. "Development of deformable mirror for spherical aberration compensation,"

Optical Data Storage

2009, 7505, 750513-6 (2009).

[134] Ichimura, I., Maeda, F., Osato, K., Yamamoto, K., and Kasami, Y. "Optical disk recording using a GaN blue-violet laser diode,"

Jpn. J. Appl. Phys

. 39, 937 (2000).

[135] Smith, C. W., Botcherby, E. J., and Wilson, T. "Resolution of oblique-plane images in sectioning microscopy,"

Opt. Express

19(3), 2662-2669.

[136] Smith, C. W., Botcherby, E. J., Booth, M. J., Jukaitis, R., and Wilson, T.

"Agitation-free multiphoton microscopy of oblique planes,"

Opt. Lett

. 36(5), 663-665

(2011).

[137] G ö bel, W. and Helmchen, F. "New angles on neuronal dendrites in vivo,"

Journal of Neurophysiology

98(6), 3770-3779 (2007).

[138] Mansoor, H., Zeng, H., Chen, K., Yu, Y., Zhao, J., and Chiao, M. "Vertical optical sectioning using a magnetically driven confocal microscanner aimed for in vivo clinical imaging,"

Opt. Express

19(25), 25161-25172 (2011).

[139] Schwertner, M., Booth, M., and Wilson, T. "Characterizing specimen induced aberrations for high NA adaptive optical microscopy,"

Opt. Express

12(26), 6540-6552

(2004).

[140] Booth, M. J., Neil, M. A. A., and Wilson, T. "Aberration correction for confocal imaging in refractive-index-mismatched media,"

Journal of Microscopy

192(2), 90-98

(1998).

[141] Booth, M. J. "Wavefront sensorless adaptive optics for large aberrations,"

Opt.

Lett.

32(1), 5-7 (2007).

[142] Rueckel, M., Mack-Bucher, J. A., & Denk, W. "Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,"

17137–17142 (2006).

PNAS

103(46),

[143] Botcherby, E. J., Smith, C. W., Kohl, M. M., DÃbarre, D., Booth, M. J.,

JuÅkaitis, R., Paulsen, O., and Wilson, T. "Aberration-free three-dimensional multiphoton imaging of neuronal activity at kHz rates,"

PNAS

109(8), 2919-2924 (2012).

201

[144] Albert, O., Sherman, L., Mourou, G., Norris, T. B., and Vdovin, G. "Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,"

Opt. Lett

. 25(1), 52-54 (2000).

[145] Poland, S. P., Wright, A. J., and Girkin, J. M. "Active focus locking in an optically sectioning microscope utilizing a deformable membrane mirror,"

Opt. Lett.

33(5), 419-421 (2008).

[146] Chang, S., Lee, J. H., Kim, S. P., Kim, H., Kim, W. J., Song, I., and Park, Y.

"Linear astigmatism of confocal off-axis reflective imaging systems and its elimination,"

Appl. Opt.

45(3), 484-488 (2006).

[147] Rogers, J. R. "Techniques and tools for obtaining symmetrical performance from tilted-component systems,"

Optical Engineering

39(7), 1776-1787 (2000).

[148] Graff, K. F.

Wave Motion in Elastic Solids

, Mineola, NY: Dover Publications

(1991).

[149] Kner, P., Sedat, J. W., Agard, D. A., and Kam, Z., "High-resolution wide-field microscopy with adaptive optics for spherical aberration correction and motionless focusing,"

J Microsc

237, 136-47 (2010).

[150] Wilson, T. "Resolution and optical sectioning in the confocal microscope,"

Journal of Microscopy

244(2), 113-121 (2011).

Download